Benford waiting-times conjecture for Mersenne numbers

Let $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, and let $P(d)$ denote the Benford probability of leading digit $d$. The waiting time between successive occurrences of a leading digit $d$ is the number of consecutive terms between those occurrences, counted as in the paper's definition of Benford-distributed waiting times. Benford waiting-times conjecture. The sequence $\{M_n\}$ has Benford-distributed waiting times: for each leading digit $d$, these waiting times behave like geometric random variables with parameter $p=P(d)$. The paper observes geometric waiting-time behavior numerically, while the corresponding local distribution properties remain conjectural.