The distinct-prime-divisor conjecture for Mersenne-type numbers

Let $n$ be a positive integer divisible by $\ell$ distinct primes, and suppose that

$$\gcd(2^n-1,n)=1.$$

Distinct-prime-divisor conjecture. If $\ell\geq 3$, then $2^n-1$ is divisible by at least $2^\ell$ distinct primes.

The preceding lemma gives only the lower bound $2^\ell-1$ in the case $p=2$; the authors state this stronger bound because they have no examples attaining $2^\ell-1$ when $\ell\geq 3$.