I think it depends on what the view is within the club. As I mentioned, if the board and management team are seeing a coach whose ideas just haven't quite clicked properly yet, or who has simply had the misfortune of being dealt the result of a mediocre transfer window, but who is still confident, has a coherent plan, and who the players are still fully behind, then I don't think even a few more losses will change that.
Sure, fans will react on gut and based on what they see in the 90 minutes, but the board is there to be more analytical and not react emotionally. They have all the information to go on, while we don't.
However, if there are signs of things cracking behind the scenes, you'd hope the people in charge would adjust their plans accordingly.
I have every faith that they would.