Let me tick the list:
Most likely false, because, as Scott Aaronson said "If $P = NP$, then the world would be a profoundly different place than we usually assume it to be."
Yes, it's The Open Problem in computational complexity theory
Yes, it's old
It's important, again quoting Scott: "because if it would be true, there would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."
It's an equality rather than a negation
