A bound on the number of distinct prime factors of Mersenne numbers

Let $p\ge 3$ be prime and set $N_p=2^p-1$. Write $\omega(N_p)$ for the number of distinct prime factors of $N_p$. The distinct-factor bound conjecture. There exists $p_0$ such that, for every prime $p>p_0$,

$$\omega(N_p)<1.36\log p.$$

This is motivated by sieve heuristics for integers whose prime factors satisfy congruence restrictions; it would imply that only finitely many primes $p$ violate the displayed bound.