BALDOCK, Circuit Judge.

A Wyoming Highway Patrol trooper stopped Defendant Lee Vang Lor for speeding in March 2007. After gaining consent to search the vehicle, the trooper found methamphetamine. The district court denied Defendant's motion to suppress the methamphetamine, and Defendant entered a conditional guilty plea to one count of possessing methamphetamine with intent to distribute and one count of conspiring to do the same. The district court sentenced him to 121 months' imprisonment. We affirmed the district court's denial of the suppression motion. United States v. Thao, 291 Fed.Appx. 129 (10th Cir.2008).

Defendant then filed a pro se petition to vacate his sentence under 28 U.S.C. § 2255 on the basis of newly discovered evidence. This new evidence was that the Wyoming Highway Patrol terminated the trooper who stopped Defendant because the trooper called in a false dispatch report in April 2007, after Defendant's arrest but before his suppression hearing. Defendant argued this evidence would have undermined the trooper's testimony at Defendant's suppression hearing and that he is therefore entitled to a new hearing. The district court denied Defendant's § 2255 petition. Defendant appealed. Exercising jurisdiction pursuant to 28 U.S.C. § 1291, we affirm the district court's denial of his petition.

I.

Wyoming Highway Patrol trooper Ben Peech stopped an SUV driven by Defendant for speeding on Interstate 80. Defendant and his passenger, Lee Thao, both said they had been to Reno, Nevada, but gave inconsistent stories as to why they were there. Trooper Peech also noted other suspicious circumstances. For example, the vehicle had been rented in Minnesota by a third party who lived in Wisconsin. Peech gave Defendant a speeding ticket, and then Thao consented to a search of the vehicle. After consenting, but before Peech began the search, Thao asked if he could have refused consent. Peech said "yes" he could "absolutely refuse." Thao then said he did not want the car to be searched. Peech acknowledged the withdrawn consent, but said he was going to detain the men pending the arrival of a drug dog. Thao and Defendant then gave verbal and written consent to a search of the SUV. The search, conducted by Peech and three other troopers, yielded approximately two pounds of crystal methamphetamine.

At the suppression hearing on May 31, 2007, Defendant and Thao testified that while the search was in progress Thao yelled at the troopers to stop. Each of the four troopers present during the search testified that neither Defendant nor Thao made any attempt to revoke their consent to search. On appeal from the district court's denial of their motion to suppress, Defendant and Thao argued (1) Trooper Peech unreasonably detained them and (2) their consent to search the vehicle was involuntary. Thao, 291 Fed.Appx. at 132. We held that Peech had reasonable suspicion to detain the men based on a number of facts. Id. at 133. As to the consent issue, we relied on the district court's credibility determinations with respect to Defendant, Thao, and the four officers who testified. Id. at 134. The district court observed in the suppression hearing, "I was more impressed with the truthfulness and testimonies of the sworn peace officers than I am of an individual who has had numerous encounters in a confrontational situation with peace officers all over and who is a convicted felon...." Id. The district court also noted it had observed both Defendant and Thao's demeanor during the testimony. Id.

Defendant outlined the factual basis for his § 2255 petition as follows:

Record on Appeal (ROA) at 12. The Government introduced the referenced newspaper article, as well as Peech's report of the stop, into the record. The article indicates Trooper Peech was cooperating with the federal Drug Enforcement Agency, which directed him to conduct a "wall stop" of a silver Dodge truck.1 ROA at 74-75. When he realized he would not be on patrol after midnight because of state cost cuts, Trooper Peech called Wyoming's drunk driving hotline and falsely reported seeing a vehicle driven by someone under the influence. Id. at 75. After being called on duty to respond to his own false report, he stopped the Dodge pickup for driving four miles per hour over the speed limit. Id. at 56. After gaining consent to search the pickup, he found $3.3 million in cash inside. Id. at 75.

Defendant argued in his § 2255 petition that because the "main issue during the suppression hearing was one of credibility," the district court "would have ruled differently in light of this new evidence." Id. at 14. The district court rejected this argument, and held that Defendant had a full opportunity to litigate his claim. The court said the "newly discovered evidence" concerning Trooper Peech was from an unrelated incident and "[a]t best ... might have provided a basis for impeaching Trooper Peech's credibility." Defendant appealed, seeking a certificate of appealability, which the district court had denied. Without granting the certificate, we ordered the Government to brief the following questions: Whether evidence discovered after a Fourth Amendment claim has been fully litigated can ever be the basis for relief under 28 U.S.C. § 2255, and if so, under what standard that evidence should be assessed. After the Government submitted its brief, we granted a certificate of appealability as to those questions, appointed Defendant counsel, and ordered counsel to file supplemental briefing. The matter has now been fully briefed.

II.

A federal prisoner is entitled to have his sentence vacated or corrected if it was "imposed in violation of the Constitution or laws of the United States ... or is otherwise subject to collateral attack...." 28 U.S.C. § 2255(a). Defendant makes two arguments in support of his § 2255 motion. First, he argues that he had no full and fair opportunity to litigate his Fourth Amendment claim because he did not have "crucial evidence needed to impeach the Government's sole witness to establish reasonable suspicion." Aplt.'s Supp. Br. at 19. Second, he argues he is entitled to an evidentiary hearing to determine whether the Government violated Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), which requires the Government to disclose material exculpatory evidence prior to trial. We take these arguments in reverse order.

A.

Defendant's second argument is that the district court erred in not conducting an evidentiary hearing to consider his Brady claim. But Defendant's § 2255 petition did not mention Brady or request an evidentiary hearing. Defendant raised Brady only in his reply brief before the district court. Accordingly, the district court did not discuss Brady, but only mentioned it in passing. In civil cases, at least, a party waives an issue in the district court if he waits to raise the argument until his reply brief. F.D.I.C. v. Noel, 177 F.3d 911, 915-16 (10th Cir.1999) (concluding the appellant waived an issue by not raising it until his reply brief in the district court). Section 2255 petitions are, admittedly, "strange and slippery creatures" that defy "traditional `civil' or `criminal' characterization." United States v. Simmonds, 111 F.3d 737, 742, 743 (10th Cir.1997), overruled on other grounds by United States v. Hurst, 322 F.3d 1256 (10th Cir.2003). But the same rule logically applies in a § 2255 proceeding, considering that plain error review does not apply to forfeited errors on § 2255 review. United States v. Frady, 456 U.S. 152, 166, 102 S.Ct. 1584, 71 L.Ed.2d 816 (1982). The Ninth Circuit reached exactly this conclusion. United States v. Berry, 624 F.3d 1031, 1039 n. 7 (9th Cir.2010) (holding that a Brady claim raised by a pro se § 2255 petitioner in his district court reply brief was too late and was not within the certificate of appealability). Of course, we construe pro se pleadings liberally, United States v. Tony, 637 F.3d 1153, 1155 n. 1 (10th Cir.2011), but even construed as liberally as possible, Defendant's § 2255 petition simply does not raise a Brady claim. Therefore, we do not consider his second argument on appeal.2

B.

Defendant's first argument is that the newly discovered evidence of Trooper Peech's deception entitled him to § 2255 relief because it shows he was "denied the opportunity for `full and fair litigation' ... of his Fourth Amendment claims in the trial court or on direct appeal." Aplt.'s Supp. Br. at 12. Defendant's petition asked the district court to grant him a new suppression hearing so he could present the newly discovered evidence.3

The Fourth Amendment protects persons against unreasonable searches and seizures. U.S. Const. amend. IV. The Supreme Court has enforced the Fourth Amendment through the exclusionary rule, a "prudential doctrine created ... to compel respect for the constitutional guaranty." Davis v. United States, ___ U.S. ___, 131 S.Ct. 2419, 2426, 180 L.Ed.2d 285 (2011) (internal quotation marks omitted). Suppression of evidence is not "a personal constitutional right," nor is it designed to "redress the injury to the privacy of the victim." Stone v. Powell, 428 U.S. 465, 486, 96 S.Ct. 3037, 49 L.Ed.2d 1067 (1976). Rather, the rule is a "prophylactic device intended generally to deter Fourth Amendment violations by law enforcement officers." Id. at 479, 96 S.Ct. 3037 (quoting Kaufman v. United States, 394 U.S. 217, 224, 89 S.Ct. 1068, 22 L.Ed.2d 227 (1969)).

The exclusionary rule "has been restricted to those areas where its remedial objectives are thought most efficaciously served." Id. at 486-87, 96 S.Ct. 3037 (quoting United States v. Calandra, 414 U.S. 338, 348, 94 S.Ct. 613, 38 L.Ed.2d 561 (1974)). Thus, the Supreme Court in Stone v. Powell declined to extend the exclusionary rule to habeas corpus proceedings by state prisoners "where the State has provided an opportunity for full and fair litigation of a Fourth Amendment claim." Id. at 494, 96 S.Ct. 3037. The Court concluded the costs of applying the exclusionary rule on collateral review outweighed the rule's deterrent effect. Id. at 493, 96 S.Ct. 3037. "The view that the deterrence of Fourth Amendment violations would be furthered rests on the dubious assumption that law enforcement authorities would fear that federal habeas review might reveal flaws in a search or seizure that went undetected at trial and on appeal." Id. And even if "some additional deterrent effect would be presented in isolated cases," the resulting protection to Fourth Amendment rights "would be outweighed by the acknowledged costs to other values vital to a rational system of criminal justice." Id. at 494, 96 S.Ct. 3037.

We have expanded the Stone bar to § 2255 petitions.4 United States v. Cook, 997 F.2d 1312, 1317 (10th Cir.1993). Thus, "Fourth Amendment violations are not reviewable in a § 2255 motion when the federal prisoner has had a full and fair opportunity to litigate the Fourth Amendment claim at trial and present issues on direct appeal." Id. at 1317.

The Stone Court did not define a "full and fair opportunity to litigate," but only said in a footnote, "Cf. Townsend v. Sain, 372 U.S. 293, 83 S.Ct. 745, 9 L.Ed.2d 770 (1963)." Stone, 428 U.S. at 494 n. 36, 96 S.Ct. 3037. This citation to Townsend is not particularly illuminating because Townsend described when a federal court in habeas corpus proceedings must hold an evidentiary hearing to develop additional facts. That question turns on whether the "the habeas applicant ... receive[d] a full and fair evidentiary hearing in a state court." Townsend, 372 U.S. at 312, 83 S.Ct. 745. Because Stone's "single, oblique footnote" referencing Townsend is not very helpful, we have concluded that "Townsend is not the sole measure of the meaning of `opportunity for full and fair litigation.'" Gamble v. Oklahoma, 583 F.2d 1161, 1164 (10th Cir.1978). Instead, we have focused on the "procedural `opportunity' aspect of the Stone standard." Id. at 1165. An "opportunity" for full and fair consideration requires at least "the procedural opportunity to raise or otherwise present a Fourth Amendment claim," a "full and fair evidentiary hearing," and "recognition and at least colorable application of the correct Fourth Amendment constitutional standards." Id. at 1165. Ineffective assistance of counsel may deny a defendant this opportunity. Cook, 997 F.2d at 1318.

Defendant identifies no procedural deficiencies in this case and does not claim his counsel was ineffective. Instead, he makes two arguments. First, he argues the existence of newly discovered evidence indicates Defendant did not have a full and fair opportunity to litigate his Fourth Amendment claim. He bases this argument on Stone's enigmatic reference to Townsend. Townsend said a federal court ought to grant an evidentiary hearing in a habeas case "[i]f ... there is a substantial allegation of newly discovered evidence." Townsend, 372 U.S. at 313, 83 S.Ct. 745. Of course, Townsend did not address a Fourth Amendment situation. In fact, it said the new evidence "must bear upon the constitutionality of the applicant's detention; the existence merely of newly discovered evidence relevant to the guilt of a state prisoner is not a ground for relief on federal habeas corpus." Id. at 317, 83 S.Ct. 745. But new evidence related to a suppression motion bears upon neither the habeas petitioner's guilt or innocence nor the constitutionality of his detention in federal custody. So we do not think the Stone Court, in citing Townsend, meant to automatically require an evidentiary hearing whenever there was "newly discovered evidence" related to a Fourth Amendment claim.

Defendant next argues the newly discovered evidence in this case establishes a Brady violation, which, unlike a Fourth Amendment violation, is cognizable on collateral review. Defendant says "If proved, [Defendant's] newly discovered evidence would establish a Brady violation that would allow the reexamination of his Fourth Amendment claim through his § 2255 motion." Aplt.'s Supp. Br. at 21. This argument goes a step too far. As we discussed above, Defendant failed to raise Brady in the district court, and he cannot bring it in through the back door. So the only way Defendant can prevail is if his lack of access to the impeaching evidence deprived him of a full and fair opportunity to litigate his Fourth Amendment claim. This is a question of law we review de novo. Miranda v. Cooper, 967 F.2d 392, 401 (10th Cir.1992).

We have never considered whether evidence discovered after a suppression hearing can circumvent the Stone bar. The Seventh Circuit, however, considered a similar situation in Brock v. United States, 573 F.3d 497, 501 (7th Cir.2009). There, the defendant brought a § 2255 motion after he located a witness who was previously unavailable to testify at the defendant's suppression hearing. Id. at 498. The witness's testimony would have potentially rendered the search warrant for the defendant's apartment invalid. Id. at 499. The court held that the later-discovered evidence did not deprive the defendant of a full and fair opportunity to litigate. Id. at 501. The court said the defendant's arguments did not "bear[] upon the central issue in Stone and the one which we must ultimately decide: whether the deterrent benefit of applying the exclusionary rule in a particular situation outweighs the social costs of letting the guilty go free, expending limited judicial resources, and disturbing finality in criminal trials." Id. at 501. The court reasoned that police were unlikely to be deterred by fear that Fourth Amendment violations undiscovered during trial and direct appeal would surface in a § 2255 motion, "at least when there is no allegation that the officers prevented the defendant from finding the evidence." Id. The "additional incremental deterrent effect" of applying the exclusionary rule to newly discovered evidence "would be outweighed by the acknowledged costs to other values vital to a rational system of criminal justice." Id. (quoting Stone, 428 U.S. at 494, 96 S.Ct. 3037). The court concluded, "Stone prevents [the defendant] from bringing this § 2255 motion solely on the ground that newly discovered evidence would have triggered the exclusionary rule if presented earlier." Id.

We agree with the Seventh Circuit's decision in Brock. A defendant is not deprived of a full and fair opportunity to litigate simply because he does not discover all potentially relevant evidence until after his suppression hearing. Under Stone, habeas corpus relief is unavailable as long as the Government afforded "an opportunity for full and fair litigation of a Fourth Amendment claim." Stone, 428 U.S. at 482, 96 S.Ct. 3037 (emphasis added). Absent ineffective assistance of counsel or government concealment, a defendant cannot claim that the mere existence of undiscovered material evidence deprived him of an opportunity to litigate his claim. As the Court said in Stone, "[W]here the State has provided an opportunity for full and fair litigation of a Fourth Amendment claim, the Constitution does not require that a state prisoner be granted federal habeas corpus relief on the ground that evidence obtained in an unconstitutional search or seizure was introduced at his trial." Stone, 428 U.S. at 482, 96 S.Ct. 3037.

Defendant says this case differs from Stone and Brock because the Government withheld the impeaching evidence. Whether or not Brady applies at the suppression stage, we can at least assume that Defendant might be deprived of a "full and fair evidentiary hearing" if the Government withholds material evidence. Cook, 997 F.2d at 1318. But nothing suggests the Government covered up evidence. The Wyoming Highway Patrol did not put Trooper Peech on paid leave until October 2007, four months after Defendant's suppression hearing. No one involved in this case, other than Peech himself, apparently knew about the false dispatch report at the time of Defendant's suppression hearing.

Defendant cites our cases indicating that "[f]or purposes of Brady, knowledge by police or investigators is imputed to the prosecution." Smith v. Sec'y of N.M. Dep't of Corr., 50 F.3d 801, 824 (10th Cir.1995) (internal quotation marks and alterations omitted). But, even if this Brady authority applies, Trooper Peech's knowledge of his own false dispatch report can hardly be imputed to the Government. The Supreme Court has explained that "the individual prosecutor has a duty to learn of any favorable evidence known to the others acting on the government's behalf in the case, including the police." Kyles v. Whitley, 514 U.S. 419, 437, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995). But Brady "does not require the prosecution to divulge every possible shred of evidence that could conceivably benefit the defendant." Smith, 50 F.3d at 823. We do not think prosecutors have a duty to investigate officers' actions in entirely unrelated cases just in case some impeaching evidence may show up. See Giglio v. United States, 405 U.S. 150, 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972) (saying prosecutors should establish procedures "to insure communication of all relevant information on each case to every lawyer who deals with it" (emphasis added)). Nor would Trooper Peech have thought he had a duty to disclose the information at the time since it was part of an unrelated, secret DEA investigation. Of course, things might have been different if the Wyoming Highway Patrol had begun investigating Peech for possible misconduct before the suppression hearing. But nothing indicates that was the case. So even assuming the Government has a Brady-like duty to disclose all material evidence prior to a suppression hearing, that duty does not extend to discovering every tidbit of information that is, or could ripen into, impeachment evidence.5

Even if the Government can be charged with knowledge of the evidence here, that would not be a sufficient reason to afford Defendant a second suppression hearing. This is because the policies behind the exclusionary rule do not support allowing this evidence to circumvent the Stone bar. The exclusionary rule is designed to "deter[] ... police conduct that violates Fourth Amendment rights." Stone, 428 U.S. at 487, 96 S.Ct. 3037. But the new evidence in this case was simply impeaching evidence that called into question Trooper Peech's credibility.6 So a new suppression hearing would only be appropriate if suppressing the evidence would deter future Fourth Amendment violations. For a number of reasons, we cannot say suppression would have this effect. First, evidence that merely impeaches an officer's credibility is not the same as direct evidence of a Fourth Amendment violation. Allowing the introduction of impeaching evidence on collateral review can only deter Fourth Amendment violations where (1) the district court's ruling on the suppression motion hinges entirely on credibility and (2) the testifying officer could foresee at the time of the alleged Fourth Amendment violation that the impeaching evidence might be discovered and damage his credibility at a second suppression hearing.

Such a situation seems highly unlikely. The Supreme Court has said the likelihood of additional impeaching evidence being discovered after the suppression hearing but before collateral review is too small to affect police officers' calculations. Id. at 493, 96 S.Ct. 3037. This is especially true when, as here, the impeaching evidence arose after the alleged Fourth Amendment violation. At the time of the allegedly nonconsensual search of Defendant's vehicle, Trooper Peech had not yet falsified the drunk driver report. We find it hard to believe officers will avoid violating the Fourth Amendment for fear that some wrongful act they might commit in the future will come back to haunt them on collateral review. Allowing impeachment evidence around the Stone bar might deter some police misconduct that is useful for impeachment, such as evidence the officer falsified reports. But the exclusionary rule is aimed only at a narrow class of police misconduct — Fourth Amendment violations. In short, allowing evidence of Trooper Peech's false report to be introduced on collateral review would do little or nothing to deter future Fourth Amendment violations. Allowing the evidence would therefore be contrary to the purposes of the exclusionary rule and inconsistent with Stone v. Powell.

For the foregoing reasons, Defendant is not entitled to § 2255 relief.

AFFIRMED.