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