What beliefs or hypotheses do you hold strongly despite not feeling confident that you can necessarily prove or evince it?

Gödel's incompleteness theorem is one of the main things that I like pointing to when talking about stuff like this, thanks for bringing it up. Its a good supporting piece that helps show there are limits to logic and knowability. I think physics models will eventually have their own version that puts theories of everything in jepoardy.

Like consciousness being greater than the sum of its parts and there being spiritual aspects to the universe. Like emotions existing as non localized complex energy frequencies, and karma existing.

I used to be a hardcore scientific determinist athiest. The scientific method, mathematical logic, and unfalsifiablility were collectively my God. My version of the universe was a mechanical box our fates predetermined by an uncaring system. There was no room for magical thinking or maybe invisible unicorns. Thing either existed or they didn't, yes or no, 1 or 0. Everything not absolute verifyable truth was worthless.

Then I had a psychedelics phase, astral projected, experienced ego death, had telepathic communications with divine / cosmicbconsciousnesses using plants as mediums, looked at myself from third person with nonexistent eyeballs, ect, ect.

I will never be able to prove to anyone my experiences are real, but what I experienced was real to me from my subjective reference frame in every way that matters.