I bought a Xiaomi A3 about four years ago. I’m still using it now but my experience has been mixed.
Two years in, my carrier told me I had to switch phones because it didn’t support a bunch of technology they now require. It in fact supported everything including all the correct GSM bands, but ATT was now operating on a whitelist to which my phone did not belong, and they would not produce an explanation as to why. I was able to switch to T-Mobile at a reduced rate, but I find the coverage leaves much to be desired by comparison.
Just after going through a few examples in my head, the difficulty becomes somewhat more apparent. let’s start with 3. This is odd, so 3(3)+1 = 10. 10 is even so we have 10/2=5.
By this point my intuition tells me that we don’t have a very obvious pattern that we can use to decide whether the function will output 4, 2, or 1 by recursively applying the function to its own output, other than the fact that every other number that we try appears to result in this pattern. We could possibly reduce the problem to whether we can guess that the function will eventually output a power of 2, but that doesn’t sound to me like it makes things much easier.
If I had no idea whether a proof existed, I would guess that it may, but that it is non-trivial. Or at least my college math courses did not prepare me to find one. Since it looks like plenty of professional mathematicians have struggled with it, I have no doubt that if a proof exists it is non-trivial.