Benford error conjecture for Mersenne numbers

Let $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime, let $D(M_n)$ denote the leading digit of $M_n$, and define the Benford error by

$$E_d(N)=\#\{n\leq N:D(M_n)=d\}-NP(d).$$

Here $P(d)$ is the Benford probability of the digit $d$. Benford error conjecture. For any fixed $\epsilon>0$, the errors satisfy

$$E_d(N)=O(N^{1/2+\epsilon})$$

and

$$E_d(N)\not=O(N^{1/2-\epsilon}).$$

The proved global Benford law gives only the limiting frequencies, whereas this conjecture predicts square-root-scale fluctuations and an essentially optimal upper bound; it remains open.