Are there infinitely many pairs of $(a, b)$ of relatively prime integers $a > 1$ and $b > 1$ such that $a^b+b^a$ is divisible by $a+b$?

I've spent almost two hours on this question to no avail. The small cases I have tried, such as $(3, 5)$, $(3, 7)$ and $(5, 7)$ all work, so I'm conjecturing that the answer is yes. However, all methods of proof I have tried have failed. Tried modulo arguments but can't really simplify my results.

Any help would be greatly appreciated.