Dynamical characterization of infinitely many Mersenne primes

Let $M(n)=2^n-1$. Let $\delta$ be the doubling map on angles, and let $k_q$ denote the period of the angle $1/q$ under $\delta$. For a prime $q$, consider primes $q$ satisfying $1<q\leq \left\lfloor\sqrt{M(n)}\right\rfloor$.

Dynamical Mersenne-prime conjecture. For every $n_0>0$, there exists a prime integer $n>n_0$ such that, for every prime $q$ with $1<q\leq \left\lfloor\sqrt{M(n)}\right\rfloor$, the angle $1/q$ has period $k_q\ne n$ under the doubling map $\delta$.

By the preceding theorem, the condition is intended as a dynamical reformulation of the existence of infinitely many prime Mersenne numbers: the relevant small prime denominators do not yield periods equal to $n$. The supplied text does not give a resolution of this formulation.