Local Benford distribution conjecture for Mersenne numbers

Let $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, and consider the leading digits of consecutive terms $M_n,M_{n+1},\ldots,M_{n+k-1}$. A digit is Benford-distributed when its probability is $P(d)$ for leading digit $d$. Local Benford distribution conjecture. For every positive integer $k$, the leading digits of $k$-tuples of consecutive terms in the sequence $\{M_n\}$ behave like $k$ independent Benford-distributed random variables. Numerical evidence supports this local independence, but no proof is given and the conjecture remains open.