I'm not sure about this.
I think he has to start showing signs of progress as opposed to just being in survival mode. If we can get say at least 6 points then I think we're in a reasonable position to push on for top 6/7. Anything less than that, then I think the pressure going into Everton, Palace and Bournemouth (who have all had good starts to the season) is immense.
As I've said before though, the narrative will be fixed for him after the Liverpool game. If that goes really badly, then he's done. That will be when the "moderates" in the ground turn.
It might not be straight away but it puts him at a point where he's one more defeat from a sacking unless he can put together a run of 5/6 wins in a row immediately after. The board will be looking for his "Watford away" game throughout that and I'm not sure he can survive that level of scrutiny for a period of time.
That being said, if we can win at Anfield, then it all completely flips. The fans are behind him. The media is talking about us being in a decent position despite having three of the hardest matches within the first part of the season.