Tao showed that proving “no Collatz cycles” is at least as hard as proving “2^l - 3^s grows fast with l and s”. The latter thing is itself very difficult, only proved in the 1960s by Fields Medal winner Alan Baker. Tao’s reasoning was covered in this post.
So, how about divergence? Is proving “no number diverges to infinity” at least as hard as some other problem Y?
I don’t mean, “if we had as-yet-unproven result Y, then we could use it to prove no number diverges”, but the converse, “if we somehow proved no number diverges, such a proof would instantly imply Y”.
This would be especially useful if Y were famously unsolved (extreme case: Riemann hypothesis), or famously solved with great effort (extreme case: Fermat’s Last Theorem).
The idea is to at least help explain why proving no-divergence seems so difficult, by identifying an tangible boulder in the road.
Following Tao’s logic, we might ponder why no-divergence seems hard, then look for a theorem Y (or conjecture Y) where we could mechanically construct, from any counter-example to Y, a divergent Collatz number. What weird thing, if true/false, would allow us to construct, locate, or prove the existence of a divergent number?
When we talk about cycles, it’s not hard to get into a technical discussion about the divisibility of certain \beta values by 2^l - 3^s being unlikely (because the latter is so big … but, hey, what if it weren’t so big?).
But when it comes to divergence, I don’t find it easy to kick off a similar discussion.