If you hit even exactly 50% of the time you will grow to infinity - that is a sequence of even/odd will grow so long as the sequence continues, it is only when the sequence isn't even/odd that you will shrink. So you need to prove that you are hitting even more than x% of the time where x > 50. I'm not sure how to calculate the exact value of X though.
This is one case where a single counter example would end the whole debate in about 5 minutes (I'm going to guess that is about how long it will take to calculate your whole sequence). However in absence of a counter example and the large number of of numbers that pass: it seems like it must be true - but we don't know how to prove it. We don't even know if it is possible to prove it (a proof that you can't prove it could be the greatest advance in math to this date - but again it is extremely hard to do that)
You would first have to prove that the number is actually a counterexample. I don't know what you mean by calculating the whole sequence, as it would be an infinite sequence. Proving a counterexample seems like a halting problem kind of problem.
But the question of a counterexample (eventhough it probably doesn't exist) is interesting. Is there a number for which we don't know weather it is a counterexample because it would take too long to compute? I feel like any single counterexample you'd propose would be proven wrong pretty quickly.
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.
This is one case where a single counter example would end the whole debate in about 5 minutes (I'm going to guess that is about how long it will take to calculate your whole sequence). However in absence of a counter example and the large number of of numbers that pass: it seems like it must be true - but we don't know how to prove it. We don't even know if it is possible to prove it (a proof that you can't prove it could be the greatest advance in math to this date - but again it is extremely hard to do that)