Yes, but as far as I know, nobody has shown that the Collatz conjecture is anything other than a really hard problem. It isn't terribly difficult to mathematically imagine that perhaps the Collatz problem space considered generally encodes Turing complete computations in some mathematically meaningful way (even when we don't explicitly construct them to be "computational"), but as far as I know that is complete conjecture. I have to imagine some non-trivial mathematical time has been spent on that conjecture, too, so that is itself a hard problem.

But there is also definitely a place where your axiom systems become self-referential in the Busy Beaver and that is a qualitative change on its own. Aaronson and some of his students have put an upper bound on it, but the only question is exactly how loose it is, rather than whether or not it is loose. The upper bound is in the hundreds, but at [1] in the 2nd-to-last paragraph Scott Aaronson expresses his opinion that the true boundary could be as low as 7, 8, or 9, rather than hundreds.

[1]: https://scottaaronson.blog/?p=8972