Does, for every prime $p$, the density $\delta_p$ of integers with $h(n)=p$ exist? Does $\liminf h(n)=\infty$? Is it true that if $p$ is the greatest prime such that $p-1\mid n$ and $p>n^\epsilon$ then $h(n)=p$?

OPEN

Let $h(n)$ be maximal such that $2^n-1,3^n-1,\ldots,h(n)^n-1$ are pairwise coprime.

Does, for every prime $p$, the density $\delta_p$ of integers with $h(n)=p$ exist? Does $\liminf h(n)=\infty$? Is it true that if $p$ is the greatest prime such that $p-1\mid n$ and $p>n^\epsilon$ then $h(n)=p$?

It is easy to see that $h(n)=n+1$ if and only if $n+1$ is prime, and that $h(n)$ is unbounded for odd $n$.

It is probably true that $h(n)=3$ for infinitely many $n$.