I find the conjecture that there are infinitely many Fermat primes to be very interesting, because there is a very reasonable-seeming and semi-quantitative heuristic argument for the conjecture, but also an equally reasonable-seeming heuristic argument against it.
Are there any other conjectures or theorems such that one can provide a short and easy-to-understand heuristic argument for the conjecture, and an equally plausible heuristic argument against it? Preferably this would be for a conjecture and heuristic arguments that are easy for a non-expert to understand, and heuristic arguments that are plausible/precise enough that someone who wasn't previously familiar with the problem would be quite confident that either argument was correct until they heard the other one. Ideally, the conjecture would still remain an open problem. (I'm not interested in a long, detailed proof that is incorrect because of some subtle logical error, of which there have of course been many examples.)
My question is somewhat related to this one.