Can a prosecutor impeach defendant by asking him about his ex-girlfriend's allegation that he was involved in a home invasion robbery?

Although the trial court tentatively ruled the evidence regarding the home-invasion robbery was not admissible, defendant's testimony on direct that he did not "join in any home robberies with [John]" made the issue relevant. Defendant's statement was broad and was not necessarily limited to the robberies John committed against people he knew. It was not improper for the prosecutor to attempt to impeach defendant by asking him about his involvement with the home-invasion robbery as described by his ex-girlfriend, especially since the prosecutor's questions were based on a good faith belief. (See People v. Friend, supra, 47 Cal.4th at p. 80 [a prosecutor may not ask questions of a witness that suggests facts harmful to a defendant if he lacks a good faith belief that such facts exist].)