Is there a positive integer $n$ such that every $n$-Mersenne prime generates a prime?

Question

First, some definitions. A $1$-Mersenne prime is just a Mersenne prime. Now, let $n$ be a positive integer. If $p$ is an $n$-Mersenne prime, and $2^p - 1$ is itself a prime, then $2^p - 1$ is said to be an $n+1$-Mersenne prime. A consequence of this definition is that, given positive integers $m$ and $n$ with $m \leq n$, if $p$ is an $n$-Mersenne prime, then $p$ is also an $m$-Mersenne prime. My question is, does there exist a positive integer $n$ such that whenever $p$ is an $n$-Mersenne prime, then $2^p - 1$ is also a prime? Now, I know that it has not yet been proven that there are infinitely many Mersenne prime numbers, but perhaps my statement can be refuted.