Mersenne Number whose Index is Mersenne Prime

Conjecture
A Mersenne number whose index is a Mersenne prime is itself a Mersenne prime.

Refutation
The smallest counterexample is $M_{8191}$.

We have that $M_{13} = 2^{13} - 1 = 8191$ is a Mersenne prime.

But $M_{8191} = 2^{8191} - 1$ is composite.