I don't think there's been any evidence to suggest that Gibby isn't already "their guy". They had the option to get rid of him many times and clearly don't want to. Atkins has already said that Gibby is part of the solution, not part of the problem.
If they get rid of Gibby, it will be to give themselves more rope. I'm sure they made it clear to Rogers from the start that this decline was going to happen and that they have a plan to compete with a smaller payroll when the big contracts are off the books.
So, TL;DR, I would be shocked if they fire Gibby. He's a good manager and a freaking great guy who has remained positive in spite of these past two years. There are no stories of any type of mutiny forming. No players are saying they are desperate to leave. If Gibby goes, that's a huge hit to my trust of Shapiro and Atkins.