The problem is if my counter example is a counter example it diverges to infinity. (someone else proposed a proof that there are no cycles other than at 1, if this proof is wrong my counter example could be a cycle and thus "easy" to show). However if goes to infinity it is really hard to see how you show it doesn't eventually converge if you just went a little big longer.

Note that every counter example you propose is actually a sequence of counter examples. It would be interesting to examine the properties of whatever numbers that counter example has in common. Though this is an obvious thing that I suspect someone has already done to no effect.