I have a question regarding Goormaghtigh conjecture on the Diophantine equation $$\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}.$$

Suppose that a positive integer $N$ is given. How many integer solutions are there to the equation $$\frac{x^m-1}{x-1}=N=\frac{y^n-1}{y-1},$$ with $x$ and $y$ prime powers?

Observe that I am not asking for a solution of Goormaghtigh conjecture in the case that $x$ and $y$ are prime powers, but I am asking whether one can bound the number of solutions with a very slow growing function of $N$, when $x$ and $y$ are prime powers. [Not sure what I mean with "slow growing". Just interested to know what is known in this case.]