KEARSE, Circuit Judge:

Defendant Amilcar Gomez appeals from a judgment of the United States District Court for the Eastern District of New York, Sterling Johnson, Jr., Judge, convicting him of racketeering conspiracy, in violation of the Racketeer Influenced and Corrupt Organizations Act ("RICO"), 18 U.S.C. § 1962(d), murder in aid of racketeering, in violation of 18 U.S.C. § 1959(a)(1), and unlawful possession of a firearm, in violation of 18 U.S.C. § 924(c)(1)(A)(i), and sentencing him principally to life imprisonment. On appeal, Gomez contends, inter alia, (1) that his Sixth Amendment right to a public trial was violated when the trial judge excluded his family during the voir dire of prospective jurors; (2) that he was denied a fair trial when the court closed the evidence without giving him an opportunity to present a defense; and (3) that he was denied his Sixth Amendment right to the effective assistance of counsel because his attorney (a) failed to object to the exclusion of his family during voir dire, (b) failed to object to the evidence-closing procedure employed by the court, and (c) failed to advise Gomez of his right to testify at trial. For the reasons that follow, we affirm the judgment of conviction. We reject all of Gomez's contentions except the claim that his attorney did not advise him of his right to testify, which we decline to reach on this appeal. That issue is more appropriate for a proceeding brought under 28 U.S.C. § 2255, in which the record can be developed in the district court.

I. BACKGROUND

The trial evidence, the sufficiency of which is not challenged, may be summarized briefly as follows. La Mara Salvatrucha ("MS-13") is a violent international street gang. Gomez was the leader of the MS-13 chapter in Jamaica, New York.

In April 2005, while Gomez was in immigration custody, he was interviewed by agents of United States Immigration and Customs Enforcement ("ICE") and, after waiving his Miranda rights, confessed to having participated in the killing of a member of a rival gang in March 2003. Gomez agreed to cooperate with ICE in its investigation of MS-13 and was released from immigration custody so that he could assist ICE proactively. In June 2005, during two meetings at the United States Attorney's Office for the Eastern District of New York ("USAO" or "Office"), Gomez again admitted participating in the March 2003 homicide, and he admitted to various other criminal activities.

In August 2006, after Gomez failed to comply with the rules set by his ICE handlers, he was taken into custody and was named in three counts of a five-count superseding indictment. He was charged with RICO conspiracy, in violation of 18 U.S.C. § 1962(d) (Count One); using and carrying a firearm during and in relation to a crime of violence, in violation of 18 U.S.C. §§ 924(c)(1)(A)(i) and 2 (Count Two); and murder in aid of racketeering, in violation of 18 U.S.C. §§ 1959(a)(1) and 2 (Count Five). Gomez was convicted on all three counts and was sentenced principally to prison terms of 365 months on Count One, to be followed by 60 months on Count Two, and, concurrently with those terms, to imprisonment for life on Count Five.

On appeal, Gomez contends principally that he was denied his right to a public trial when the court excluded his family from the courtroom during jury selection, see Part II.A. below; that he was denied the right to testify at trial and present a defense, see Part II.B. below; and that he was denied effective assistance of counsel, see Part III below. He also complains that the district court allowed the government to treat him unfairly with respect to a proffer agreement entered into in October 2005, see Part II.C. below.

II. CHALLENGES TO THE CONVICTION AND SENTENCE

A. The Right to a Public Trial

Gomez's trial was scheduled to begin on March 30, 2009. On that day, with Gomez present, his attorney requested an adjournment, stating that Gomez wished to speak with additional witnesses. The court denied the application, and the following colloquy ensued:

(Trial Transcript ("Tr.") at 3-6 (emphasis added).)

Gomez contends that this exclusion of his family from the courtroom violated his right to a public trial and that he is therefore entitled to a new trial. We disagree.

The Sixth Amendment to the Constitution grants a criminal defendant the right to a public trial, see, e.g., Waller v. Georgia, 467 U.S. 39, 43-47, 104 S.Ct. 2210, 81 L.Ed.2d 31 (1984), which "extends to the voir dire of prospective jurors," Presley v. Georgia, 558 U.S. 209, 130 S.Ct. 721, 724, 175 L.Ed.2d 675 (2010).

Waller, 467 U.S. at 46, 104 S.Ct. 2210 (quoting Gannett Co. v. DePasquale, 443 U.S. 368, 380, 99 S.Ct. 2898, 61 L.Ed.2d 608 (1979) (quoting In re Oliver, 333 U.S. 257, 270 n. 25, 68 S.Ct. 499, 92 L.Ed. 682 (1948) (other internal quotation marks omitted))); see, e.g., Morales v. United States, 635 F.3d 39, 44 & n. 11 (2d Cir. 2011) (values "generally understood to be... protected by the Sixth Amendment right to a public trial" include "(1) ensur[ing] a fair trial, (2) remind[ing] the prosecutor and judge of their responsibility to the accused and the importance of their functions, (3) encourag[ing] witnesses to come forward, and (4) discourag[ing] perjury"). And "[w]hile a public presence will more likely bring to light any errors that do occur, it is the openness of the proceeding itself, regardless of what actually transpires, that imparts `the appearance of fairness so essential to public confidence in the system' as a whole." United States v. Gupta, 699 F.3d 682, 689 (2d Cir.2012) ("Gupta")(quoting Press-Enterprise Co. v. Superior Court of California, 464 U.S. 501, 508, 104 S.Ct. 819, 78 L.Ed.2d 629 (1984)).

However, the right to a public trial "is not absolute." Gupta, 699 F.3d at 687. The right

Gupta, 699 F.3d at 687 (emphases in Gupta).

A defendant claiming deprivation of the right to a public trial need not demonstrate specific prejudice in order to obtain relief, for a violation of that right is a structural error that is not subject to harmless-error review. See, e.g., Johnson v. United States, 520 U.S. 461, 468-69, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997); Waller, 467 U.S. at 49-50, 104 S.Ct. 2210. "Whether an error can be found harmless," however, "is simply a different question from whether it can be subjected to plain-error review." Puckett v. United States, 556 U.S. 129, 139, 129 S.Ct. 1423, 173 L.Ed.2d 266 (2009).

Levine v. United States, 362 U.S. 610, 619-20, 80 S.Ct. 1038, 4 L.Ed.2d 989 (1960) (emphasis added); see also Brown v. Kuhlmann, 142 F.3d 529, 542 (2d Cir.1998) (setting aside a conviction in order to deter violations of the Public Trial Clause "serves little purpose when the defendant" has made "no more than a perfunctory effort to prevent the closure").

Puckett, 556 U.S. at 134, 129 S.Ct. 1423 (quoting Wainwright v. Sykes, 433 U.S. 72, 89, 97 S.Ct. 2497, 53 L.Ed.2d 594 (1977) (emphases ours)).

An error that was not objected to at trial may generally be reviewed under the plain-error standard. See Fed.R.Crim.P. 52(b). Under that standard, "before an appellate court can correct an error not raised at trial, there must be (1) `error,' (2) that is `plain,' and (3) that `affect[s] substantial rights.'" Johnson, 520 U.S. at 466-67, 117 S.Ct. 1544 (quoting United States v. Olano, 507 U.S. 725, 732, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993)); see Puckett, 556 U.S. at 135, 129 S.Ct. 1423 ("error" means "some sort of [d]eviation from a legal rule[] that has not been intentionally relinquished or abandoned, i.e., affirmatively waived, by the appellant" (internal quotation marks omitted) (emphasis ours)). "If all three conditions are met, an appellate court may then exercise its discretion to notice a forfeited error, but only if (4) the error `seriously affect[s] the fairness, integrity, or public reputation of judicial proceedings.'" Johnson, 520 U.S. at 467, 117 S.Ct. 1544 (quoting Olano, 507 U.S. at 732, 113 S.Ct. 1770 (other internal quotation marks omitted)). "Meeting all four prongs" of this test "is difficult, as it should be." Puckett, 556 U.S. at 135, 129 S.Ct. 1423 (internal quotation marks omitted).

Moreover, "[w]e have been especially reluctant to reverse for plain error when it is `invited.'" United States v. Young, 745 F.2d 733, 752 (2d Cir.1984) (other internal quotation marks omitted); see generally United States v. Wells, 519 U.S. 482, 488, 117 S.Ct. 921, 137 L.Ed.2d 107 (1997) ("under the invited error doctrine ... a party may not complain on appeal of errors that he himself invited or provoked" the district court to commit (internal quotation marks omitted)).

In the present case, there was no objection to the exclusion of Gomez's family during the voir dire. Rather, as the colloquy quoted above reveals, Gomez's attorney fully acquiesced in the exclusion of the family. Indeed, in response to the court's simple inquiry as to who the seated persons were, counsel immediately stated that he would tell the family to leave the courtroom for the course of the voir dire.

We think it clear from that inquiry by the district judge, when the prospective jurors were about to be brought to the courtroom, that the judge anticipated having at least some of the persons already seated leave because, as the record reveals, there were not enough seats for all of the venirepersons (see, e.g., Tr. 10 (the court, after summoning some prospective jurors to the jury box, instructed, "Those of you who are standing, you can get some of these empty seats as the jurors are called.")). Had there been a timely objection by Gomez to the exclusion of his family members, the trial court might well have adopted an alternative such as "dividing the jury venire panel to reduce courtroom congestion," Presley, 130 S.Ct. at 725. And had there been an objection, the court would have been alerted to announce the findings — required in the normal case, where there either is an objection, see, e.g., Waller, 467 U.S. at 42, 48, 104 S.Ct. 2210 (Sixth Amendment objection by the defendant); Press-Enterprise, 464 U.S. at 512-13, 104 S.Ct. 819 (First Amendment objection by the press), or the defense is unaware of the exclusion, see, e.g., Gupta, 699 F.3d at 687, 689-90 — as to the necessity for, the needed breadth of, and the possible alternatives to, any exclusion. Instead of objecting, however, Gomez's attorney volunteered to tell Gomez's family to leave.

Accordingly, even if the exclusion of the Gomez family members during the voir dire in this case was error, it cannot be viewed as one that affected the fairness, integrity, or public reputation of judicial proceedings. To the contrary, the fairness and public reputation of the proceeding would be called into serious question if a defendant were allowed to gain a new trial on the basis of the very procedure he had invited. Gomez's public trial claim provides no basis for overturning his conviction.

B. The Closing of the Evidence

After the government's last witness had testified, the jury was sent out of the courtroom, and the district court anticipated that Gomez would immediately move under Rule 29 of the Federal Rules of Criminal Procedure for a judgment of acquittal. However, the government stated that it had one additional exhibit to read to the jury before its case was complete. Gomez's counsel suggested that he could nonetheless move for acquittal immediately and that the court could simply reserve decision until the government rested. The court decided instead that the government should, initially, rest immediately and that defense counsel should make a Rule 29 motion and receive a ruling; that the government should then reopen its case and rest a second time; and that defense counsel should then make a second motion for acquittal, on which the court would rule.

After denying Gomez's first Rule 29 motion, and before bringing the jury back into the courtroom to allow the government to reopen its case, the court asked the parties about the estimated lengths of their summations, and there followed a discussion of scheduling. (See Tr. 642-43.) In the course of that discussion, Gomez's attorney indicated that he would like to take a break to confirm with Gomez that the defense did not wish to present any evidence:

(Tr. 643 (emphases added).)

Following the government's limited reopening of its case, the government rested, Gomez's attorney moved again for acquittal, and the district court denied the renewed motion. (See id. at 647-48.) The parties were then allowed to give their respective summations (see id. at 648, 673), and the case was submitted to the jury.

Gomez contends that the trial court, in adopting the above procedure, and in allowing summations to be commenced without inquiring whether the defense rested and without announcing that both sides rested or that the evidence was closed, prevented Gomez from presenting a defense. (See, e.g., Gomez brief on appeal at 28, 32.) Again, we disagree.

A defendant in a criminal trial of course has a constitutional right to present a defense, see, e.g., California v. Trombetta, 467 U.S. 479, 485, 104 S.Ct. 2528, 81 L.Ed.2d 413 (1984); see also Fed. R.Crim.P. 29(a) ("[i]f the court denies a motion for a judgment of acquittal at the close of the government's evidence, the defendant may offer evidence without having reserved the right to do so"), and to testify at trial, see, e.g., Rock v. Arkansas, 483 U.S. 44, 49-53, 107 S.Ct. 2704, 97 L.Ed.2d 37 (1987). However, if it appears that the trial judge has erroneously assumed that the defendant does not wish to present evidence, it is incumbent on the defense to voice an objection alerting the court to that error. In the absence of such an objection, any claim on appeal that the defendant was denied that right is subject to, at best, plain-error analysis.

We see no error here, much less one that is "plain" or that affected Gomez's substantial rights or the fairness, integrity, or reputation of judicial proceedings. There is no indication whatever in the record that Gomez wanted to present a defense. Rather, the trial transcript makes it clear that, after the government's last witness had testified and been excused, the court was attempting to determine an appropriate schedule for the remainder of the proceedings, and that in the course of those discussions, Gomez's attorney stated explicitly that he wished an opportunity to speak with Gomez and to "confirm that the defense is going to rest" (Tr. 643). The transcript states that there followed a discussion "[o]ff the record" (id.); and of course whatever conversation occurred between Gomez and his attorney would have been confidential. But given defense counsel's express mission of conferring with Gomez to determine whether Gomez wanted to present any evidence, the record's ensuing silence as to any such desire implies that counsel had indeed "confirm[ed] that the defense [wanted] to rest" (id.).

Accordingly, the record provides no basis for an inference that the absence of a formal declaration by counsel that the defense rested, or the absence of a formal announcement by the court that both sides had rested and that the evidence was closed, had the effect of depriving Gomez of his right to present a defense.

C. Other Contentions

Gomez also contends that the district court erred by allowing the government to introduce certain evidence at trial, allegedly in breach of an agreement between himself and the government and/or in violation of the Federal Rules of Evidence, and by refusing to compel the government to move for a reduction in his sentence on the basis of his cooperation with the government.

1. The Government's Alleged Breach of the Proffer Agreement

During the approximately one-year period prior to his arrest, when Gomez was engaged in his attempt to cooperate with ICE, he attended a series of meetings with ICE agents and with attorneys in the USAO. In October 2005, Gomez, represented by counsel, and the government entered into a proffer agreement (the "Proffer Agreement" or "Agreement"). Gomez contends that the government's introduction at trial of statements he made in a March 2006 telephone conversation with ICE Special Agent Erin Keegan, as well as the testimony of an informant located by the government as a result of Gomez's statements in that telephone conversation, violated the Proffer Agreement. (See Gomez brief on appeal at 43-49.) These contentions are meritless.

The admission of Gomez's statements to Keegan did not violate the Proffer Agreement because although the government agreed "not [to] offer in evidence any statements made by [Gomez] at the Meeting[s]" (Proffer Agreement at 1, ¶ 2 (emphasis added)), the Agreement provided that "this agreement is limited to the statements made by [Gomez] at the Meeting[s] and does not apply to any oral, written or recorded statements made by [Gomez] at any other time" (id. at 2, ¶ 6 (emphases added)). The objected-to statements by Gomez were made in a telephone call, not at a meeting.

Nor was the Proffer Agreement violated by the admission of the testimony of the informant identified by Gomez in his telephone conversation with Keegan. The Agreement provided that the government "may use any statements made by [Gomez]... to obtain leads to other evidence, which evidence may be used by the Office in any stage of a criminal prosecution." (Proffer Agreement at 1, ¶ 3 (emphases added).)

2. The Government's Alleged Violation of Rule 410

Gomez also contends that the admission of his telephone statements to Keegan violated Rule 410 of the Federal Rules of Evidence. That Rule, at the time of Gomez's trial in 2009, provided in pertinent part that "evidence of ... any statement made in the course of plea discussions with an attorney for the prosecuting authority which do not result in a plea of guilty" "is not ... admissible against the defendant who ... was a participant in the plea discussions." Fed.R.Evid. 410(4) (1980) (emphases added). This Rule would apply to statements made by telephone as well as at a meeting, so long as the telephone conversation was "in the course of plea discussions with an attorney for the prosecuting authority," id. "Whether a party [wa]s engaged in plea discussions is a factual question that must be determined on a case-by-case basis." United States v. Serna, 799 F.2d 842, 848 (2d Cir.1986), overruled on other grounds by United States v. DiNapoli, 8 F.3d 909 (2d Cir. 1993) (en banc).

Gomez has failed in at least two respects to show that his telephone conversation with Keegan was within the scope of Rule 410. First, Keegan was an ICE agent, and — even assuming arguendo that the "prosecuting authority," within the meaning of Rule 410, was ICE rather than the United States Attorney's Office — there was no evidence that Keegan was an attorney. Second, Gomez has not shown that the telephone conversation was in "the course of plea discussions." Although his attorney argued to the district court that Gomez "was cooperating at th[at] time" (Tr. 457), "the course of plea discussions" is not necessarily congruent with a period of cooperation. There was no evidence of any plea discussion in that telephone conversation; rather, Gomez's attorney stated that the conversation occurred because Gomez "called the police" "about [a] situation" (id.). Thus, Gomez has not carried his burden of showing that his telephone conversation came within Rule 410. We see no abuse of discretion in the trial court's admission of the challenged evidence.

3. The Government's Refusal To Move for a Sentence Reduction

Relying on the Proffer Agreement, Gomez contends that the district court erred in declining to compel the government to file a motion pursuant to 18 U.S.C. § 3553(e) and § 5K1.1 of the Sentencing Guidelines ("Guidelines") in order to permit the court to sentence Gomez to prison terms below, respectively, the statutory mandatory minima and the advisory-Guidelines-recommended range. Gomez contends that he was entitled to such a motion because he rendered substantial assistance to ICE, and that the government's refusal to make such motions was retaliation for his decision to proceed to trial. We reject this contention.

"[W]hen a defendant has substantially assisted" law enforcement efforts, the government has "a power," but "not a duty," to file a § 3553(e) or a § 5K1.1 motion for a sentence reduction. Wade v. United States, 504 U.S. 181, 185, 112 S.Ct. 1840, 118 L.Ed.2d 524 (1992). However, "a prosecutor's discretion when exercising that power is subject to constitutional limitations that district courts can enforce." Id. Thus, even in the absence of a cooperation agreement, "a defendant would be entitled to relief if a prosecutor refused to file a substantial-assistance motion, say, because of the defendant's race or religion." Id. at 186, 112 S.Ct. 1840; see, e.g., United States v. Khoury, 62 F.3d 1138, 1142 (9th Cir.1995) (evidence that the government withdrew its § 5K1.1 motion because Khoury "forced [the government] to go to the trouble of proving [its] case before a jury, as was his constitutional right," constitutes "the `substantial threshold showing' required by Wade"); United States v. Paramo, 998 F.2d 1212, 1214 (3d Cir.1993) ("district court has authority to grant a downward departure for substantial assistance if the government's sole motive for withholding a 5K1.1 motion was to penalize the defendant for exercising his constitutional right to a trial"). In order to warrant even a hearing with respect to the government's failure to make such a motion, however, the defendant must present more than "generalized allegations of improper motive"; he must make a "substantial threshold showing" of such a motive. Wade, 504 U.S. at 186, 112 S.Ct. 1840.

In the present case Gomez has made no such showing. The government had not agreed to make a motion on his behalf. His Proffer Agreement with the government stated that "THIS IS NOT A COOPERATION AGREEMENT" (Proffer Agreement at 1, ¶ 1 (emphasis in original)) and that in "receiving [Gomez's] proffer, the Office does not agree to ... make a motion on [Gomez's] behalf" (id. (emphasis added)). Further, Gomez's contention that the government's motive in not moving for a reduction in his case was retaliation for his insistence on exercising his right to a trial is entirely conclusory. The government, in contrast, presented the district court with concrete reasons for declining to bring such a motion. The government stated that Gomez had failed to comply with ICE instructions that he was to, inter alia, inform ICE of any plans to attend gang meetings or interact with gang members. The government also noted that, during proffer sessions, Gomez had repeatedly lied, rendering himself useless as a witness against other targets of the investigation. The district court did not err in declining to compel the government to file a § 3553(e) or a § 5K1.1 motion.

III. THE CLAIMS OF INEFFECTIVE ASSISTANCE OF COUNSEL

Gomez contends that he was denied his Sixth Amendment right to the effective assistance of counsel because his attorney (a) failed to object to the exclusion of his family during voir dire, (b) failed to object to the evidence-closing procedure adopted by the court, and (c) failed to advise Gomez of his right to testify at trial. In order to prevail on an ineffective-assistance-of-counsel ("IAC") claim, a defendant (1) "must show that counsel's performance was deficient," Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984), so deficient that, "in light of all the circumstances, the identified acts or omissions were outside the wide range of professionally competent assistance," id. at 690, 104 S.Ct. 2052; and (2) must show "that the deficient performance prejudiced the defense," id. at 687, 104 S.Ct. 2052, in the sense that "there is a reasonable probability that, but for counsel's unprofessional errors, the result of the proceeding would have been different," id. at 694, 104 S.Ct. 2052. A defendant's failure to meet either prong of this test requires rejection of his IAC claim. See, e.g., id. at 687, 697, 104 S.Ct. 2052; Bennett v. United States, 663 F.3d 71, 84-85 (2d Cir.2011); Chang v. United States, 250 F.3d 79, 84 (2d Cir.2001); Brown v. Artuz, 124 F.3d 73, 80 (2d Cir.1997).

We do not reach, on this appeal, Gomez's claim that his attorney failed to advise him of his right to testify. "Every criminal defendant," of course, "is privileged to testify in his own defense, or to refuse to do so." Harris v. New York, 401 U.S. 222, 225, 91 S.Ct. 643, 28 L.Ed.2d 1 (1971). And it is well established both (a) that "counsel must inform the defendant that the ultimate decision" whether or not to testify "belongs to the defendant, and [(b) that] counsel must abide by the defendant's decision on this matter." Brown v. Artuz, 124 F.3d at 79. However, the record with regard to whether or not Gomez desired to testify, as well as with regard to the advice he received from counsel as to both his right to testify and his right to decide whether to do so, is entirely undeveloped. Accordingly, this aspect of his contention that his attorney rendered ineffective assistance is inappropriate for resolution at this time. Gomez may bring a motion under 28 U.S.C. § 2255 in order to pursue this claim.

To the extent that Gomez makes an ineffectiveness claim based on the court's closing-of-the-evidence procedure, we reject it because Gomez fails to meet the performance prong of the Strickland standard. As a matter of procedure, we see no deficiency in counsel's failure to make a formal statement that the defense rested or to insist on a formal announcement by the court that the evidence was closed. However, our rejection of this contention as an independent IAC claim is not meant to foreclose Gomez from showing, on a § 2255 motion with respect to his right to testify, that counsel's consent to the closing of the evidence overrode a desire by Gomez to testify.

Insofar as Gomez makes an ineffectiveness claim based on his attorney's failure to object to the family's exclusion from the courtroom during jury selection, we reject it because Gomez has failed to meet the prejudice prong of the Strickland test. The arguments presented in his briefs on appeal provide no indication that if Gomez's family had been present for the voir dire, there is any reasonable probability that the result of the proceeding would have been different.

CONCLUSION

We have considered all of Gomez's contentions on this appeal, and as to those issues for which the record is sufficiently developed, we have found his contentions to be without merit. The judgment of the district court is affirmed, without prejudice to Gomez's filing a motion pursuant to 28 U.S.C. § 2255 to pursue his claim that his attorney did not advise him of his right to testify, as indicated in Part III above.