OPEN

Let $p$ be a prime. Is it true that the equation
\[(p-1)!+a^{p-1}=p^k\]
has only finitely many solutions?

Erdős and Graham remark that it is probably true that in general $(p-1)!+a^{p-1}$ is rarely a power at all (although this can happen, for example $6!+2^6=28^2$).