Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a program which will eventually output either Yes or No (or whatever sort of answer, out of finitely many possibilities, is appropriate), though it may take an implausibly long time to do so) but which remain open?

Dropping the qualifier "significant", examples of this sort of thing would be determining whether chess between perfect players results in a white win, black win, or stalemate; determining the $10^{10^{100}}$th decimal digit of $\pi$; etc. But none of these are of particular significance in mathematics, such that anyone would ordinarily list them as an open problem of note.

So, though it is inherently a subjective judgment: Are there any good examples of significant open problems of this sort?