Conjecture on the cycle counts of Mersenne numbers and their factors

Let $M_p=2^p-1$ be the Mersenne number associated with a prime number $p$. For a positive integer $q$, let $G_q/R_q$ denote the cycle decomposition of the multiplication-by-$2$ map modulo $q$, and let the number of $k$-cycles of $q$ mean the number of cycles of length $k$ in this decomposition. Mersenne-cycle-count conjecture. For each prime number $p$, the number of $k$-cycles of $M_p$ is

$$\frac{(p-1)!}{k!(p-k)!}.$$

Moreover, a positive integer $d$ other than $1$ is a factor of $M_p$ if and only if the number of $k$-cycles of $d$ is equal to the number of $(p-k)$-cycles of $d$. The conjecture is motivated by the displayed cycle decompositions and binomial expansion in the source; no proof or resolution is supplied there.