I went down the same line of thought as you when this problem nerd sniped me 5 years ago (and periodically over the last 5 years). I actually now think you may

I went down the same line of thought as you when this problem nerd sniped me 5 years ago (and periodically over the last 5 years). I actually now think you may be able to do a computer proof that there are no trivial cycles other than 1 with a clever construction of large integers that lets you consider them in chunks, but I don't think there's any chance of finding a cycle with computation absent some weird number theory. Edit: The cycle length seems to have gotten longer since I last checked.