Conjecture on highly-composite Mersenne indices

Let $N$ be an index of a highly-composite Mersenne number if $\tau(2^N-1)>\tau(2^m-1)$ for every integer $m$ with $1\leqslant m<N$. Highly-composite Mersenne index conjecture. If $N$ is restricted to indices of highly-composite Mersenne numbers, then

$$\lim_{N\to\infty}\frac{\tau(2^N+1)}{N}=\infty.$$

The conjecture proposes a growth property along record-setting Mersenne indices and is presented as a reasonable conjecture; no resolution is given in the supplied text.