I’m a little out of my depth, but I’d guess a lot of them would probably fall into one of two categories: Something we believe should go on forever (and not halt) if the math problem is resolved the way we expect, but theoretically could suddenly halt after some absurdly long number of steps. Or something where it halts for a given input after some number of steps unless something some counter example exists where it goes on forever.

In the first, you can’t really do anything but just keep watching it not halt but it isn’t telling you anything about the infinity to go. (Say a program that spits out twin primes, we expect an infinite number but we don’t really know)

And in the second case we’d just have to keep trying larger and larger inputs making this just an extension of the first category if we wrote a program to do that for us. And if we did find an example where it goes on forever without repeating states, how would you even know? It’d be like the first situation again.