Are there any solutions to $\frac{3^n - 2^n}{2^k-3^n} = N$ for $n$, $k$, $N$ $\in\mathbb{N}$, greater than 2.

This is related to a previous answered question: Are there any solutions to $2^n-3^m=1$

Things I already know:

There are no appropriate solutions to $2^k-3^n = 1$ which would make this trivial.

There is an equivalent question which is $\frac{2^{k-n} - 1}{2^k-3^n} = M$.

Things I've tried:

Several variants on diophantine equations. I've not yet found anything that involves quotients, etc. any pointers here would be welcome.

Several things to do with modulus/cyclic groups. The best outcome of this has been the above alternate form which doesn't seem any closer really.

Experimentally, there are no solutions with $n, k < 1000$. There are some pretty tight limits on what k can be as you need $2^k-3^n < 3^n - 2^n$ but also $2^k > 3^n$ which is only one or two $k$'s for every $n$. Also so far I've not seen any $N > 16$ which is a bit puzzling given the size of the values in the fraction. This makes me think there may be a limit based argument. Any pointers to where to start here would also be welcome.