Is it permissible to pray behind two imams who differ to such an extent that one leaves the mosque when the other enters?

The prayer behind a transgressor (fasiq) imam is permissible according to the majority of scholars, as long as his own prayer is valid, especially if his transgression is not related to the prayer itself. Disagreement between two imams does not affect the validity of praying behind them, but they should be advised to exemplify good conduct.