The equality for the period of a Mersenne-indexed quotient ring

Let $R_n$ be the quotient ring and let $t(n)$ denote the order of $X$ in its multiplicative structure, as in the preceding discussion. For $n\geq 2$, the period at the Mersenne index $2^n-1$ is conjectured to satisfy

Mersenne-period conjecture. For all $n\geq 2$,

$$t(2^n-1)=2^{t(n)}-1.$$

This conjecture is supported by numerical computations for $n\leq 16$ using GAP and Pari-GP; no resolution beyond those computations is given in the source.