. While this struggle was occurring, Pérez pulled his mother back inside his vehicle.

. Only the witnesses relevant to this appeal will be discussed.

. Sánchez was indeed providing information to the government. Officers inspecting Sánchez’s body found a piece of paper containing telephone numbers belonging to federal agents. One of those numbers belonged to FBI Task Force Investigative Agent Pablo Iri-zarry-Ayala ("Irizarry”), who testified that he had met with Sánchez and she' had provided the FBI with information related to both Jim-énez — who she described as the leader of an organization in the Falín Torrech housing project in Bayamón — and Alexis.

. The district court stated that

I offered you to have the witness sit here and be voir dired and be asked whether in light of that information, he would still be willing to give his testimony. And I bet you that he would have said no, because once he gets to know the reality of the facts, no serious individual is going to take the stand and give the testimony of the kind that you want under these circumstances.

It then clarified that while Jiménez’s counsel could argue "whatever [he] want[ed],” he could "not use an expert to give an imprimatur of expertise on something that [counsel] know[s] is totally false."

. In no way are we suggesting that a defendant’s proffer and admission of guilt should be taken with a grain of salt. In the run-of-the-mill case, it is in a defendant’s best interest to tell the truth, and he or she often has little incentive to lie. However, as the Supreme Court has stated for over forty years, "death is different.” See, e.g., Ring v. Arizona, 536 U.S. 584, 605-06, 122 S.Ct. 2428, 153 L.Ed.2d 556 (2002); Murray v. Carrier, 477 U.S. 478, 526, 106 S.Ct. 2639, 91 L.Ed.2d 397 (1986) (Brennan, J., dissenting); Gregg v. Georgia, 428 U.S. 153, 188, 96 S.Ct. 2909, 49 L.Ed.2d 859 (1976). A defendant facing the death penalty has a strong incentive to say whatever is needed to eliminate a potential death sentence and preserve his life. Indeed, when we asked learned counsel at oral argument whether he believes defendants sometimes admit guilt in a proffer in order to avoid *18 a severe sentence even though they are not guilty, learned counsel unequivocally and succinctly stated, "Yes.''

. To the extent one tries to distinguish this example by arguing that the expert testimony here is less compelling, we note that such an argument would hinge on a judgment about the persuasiveness of that testimony — a judgment which would go well beyond any gate-keeping role.

. We also take issue with the district court’s comment that it ''ha[s] an obligation to make certain that the facts that come out are as truthful as possible to the reality of the case. I cannot close my eyes to that reality. It would be improper, wrong for me to do that, and I will not allow that.” District courts "close their eyes” to pertinent evidence all the time. For example, that is the whole point of motions to suppress; if evidence or statements are suppressed, courts and parties pretend that the evidence does not exist. Similarly, if evidence is excluded under Rule 403 of the Federal Rules of Evidence, pertinent and relevant evidence is ignored by the court and the parties because of a belief that it is unduly prejudicial. This is no different. Jim-énez’s proffer, for all intents and purposes, did not exist unless he testified. Just like excluded evidence, the district court had an obligation to “close [its] eye to that reality” unless Jiménez took the stand.

. As we noted in Melvin, "[i]t is open to legitimate question whether the rule demanding 'automatic reversal’ based on 'policy interest[s]’ might apply” to the violation of an immunity agreement. 730 F.3d at 38 n. 3 (second alteration i