Good question. I think if Poch goes there, at the present time, Levy's in a world of trouble with a lot of the fan base. The preferred option for 50%-60% of the fan base (based on the polls I see here and elsewhere) goes to one of our two most hated rivals - that would be enough for some to get the pitchforks out. It'd also appear that Chelsea, who are chaotic to say the least, are more organised than us. Again, that just ramps the pressure up on Levy because the optics are terrible. I don't think we'll get 4th and I can see results going south to be honest. That happens and the argument can be made that "Conte had us 4th, we sacked him with no plan and we are tumbling down the table". It's a flawed argument but when tensions run high, no one cares how flawed the argument is if it suits a narrative. It'll all create toxicity around the club with Levy the target for most of it.
Levy's already under pressure IMO, Poch goes there and that ramps up. Poch is successful there, especially if the wheels come off for us, and it could turn ugly.
How much influence and Levy Out movement can have is debatable. But it certainly won't get smaller if the above comes to pass and that's not good news for our owners.