It will be up to Aaron Rodgers. If he wants Hackett, the guy who helped get him back to back MVP seasons, then he stays. I don't agree with firing Saleh, but the same applies to him.
The Jets are unable to pivot to a new coaching staff, nor a new GM, in such little time. As an organization, they seem to move slow as molasses when it comes to shuffling personnel. In addition, unloading the current less than welcoming situation to a competent and willing taker is highly unlikely. Assuming they are safe for now, if Saleh and JD are unable to construct a competitive product for '24, the likely outcome would be a new broom sweeping clean.