Are there any non-trivial solutions of $a^n = b^m + 1$ for natural $a$, $b$ and $n, m \ge 2$

Question

There is an obvious solution $3^2 = 2^3 + 1$.

However, I tried to search for solutions in the range $a, b \in [2,100]$ and $\min(n,m) \in [2,1000]$, but did not find any.

It is easy to show that $a$ and $b$ have to be co-prime, but I couldn't advance anywhere further.