I'm going to roll back a stance I've had based on some new information I've learned. I definitely think there is a 0% chance that JR gets fired, but I think there is a small, but real, chance that Sullivan gets fired. The big reason for that? His contract status,
he signed a 3 year contract that started after the 2017 cup. That means his contract is done after next season, which is a big factor to consider. Teams usually don't like having "lame duck" coaches, most teams re-sign their coaches before the last years of their contracts. Do you see them giving Sullivan an extension after these playoffs?
I think there is a small chance he gets fired this off-season, but a much bigger chance that he's not retained after next season or fired during next season. I think he'd only get fired if they have a good alternative to Sullivan available, such as if John Cooper gets fired (he won't get fired, his extension hasn't even kicked in yet). They would use his contract status more than the on ice results to justify it. I doubt they'd do that unless they can get someone like Cooper or Laviolette, though. You have to remember that despite Sullivan's flaws, his system is pretty much perfect for the Crosby/Malkin/Letang/Kessel core.
With that being said, Martin should 100% be fired and Recchi should have never been hired in the first place. I blame Martin for a majority of this team's defensive issues and Recchi is horribly unqualified to be the forwards coach.