My thought is either Crystal Palace or West Ham.
04/07 - Bournemouth (H)
09/07 - Aston Villa (A)
13/07 - Southampton (H)
19/07 - Chelsea (FA Cup)
26/07 - Leicester (A)
Some fixtures need to be rearranged due to that FA Cup tie. I am going to assume that we play Palace on 16/07 & West Ham on 22/07 or 23/07. The first three looked spaced apart quite nicely and I fancy us to win those three in a row and be in the top four by that point, defending a lead rather than chasing teams down. However, we'll want to play a strong team against Chelsea (definitely some changes for both clubs I think) with games tightly scheduled around it. So, I think one of those games is likely to be when the draw(s) or loss come.
I'd have preferred it if the FA could do us a bit of a favour and move Villa to 08/07, Southampton to 12/07 and kept Palace as 15/07 but I highly doubt they are going to move that many fixtures. At least Southampton, Palace, and West Ham could be effectively playing for nothing. So they might be quite open games. Biggest question mark is West Ham who might still need something, but I reckon Villa & Bournemouth would be too far gone by then so the Hammers won't be fighting tooth & nail against us like they might have.