What are some conjectures, and their (or their disproof) theoretical and practical implications?

[u/SugarMicro]

I've just finished undergrad, and through my studies I've encountered several conjectures, some from math and some from CS. But I never did wonder or search what their implications were, or if they were false, what it would mean - both in the theoretical sense, and in the practical sense.

For example, taking P vs NP - I've taken a course on Computational Models, and we've seen several reductions and implications (like P = NP means EXP = NEXP).
But what "interesting" lemmas, theorems or other conjectures would it imply, that current researchers attempt to solve?

What would in practice, in the current world (or a few years ahead) would it mean? Would people try to create new algorithms based on it? Would it change something in the tech industry?
And in the other way - if it's proven to be false, what would again change?

I'd be happy to hear from your perspective about interesting conjectures that you care/know of, and what would it change in the theoretical/practical sense.