Major changes needed to paper "in press"
Well, you have to fix it, don't you? Talk to the editor. 

Yes, the paper may indeed be rejected, but we cannot know this. 

As a reviewer, I have found subtle holes in papers, and really it would have been unwise for the authors to submit without taking care of filling the holes (which they thought they could do, but turned out was impossible, as counterexamples were published later). 

TL;DR: Better a watered-down, correct result than a wrong one.