Density and liminf of $h(n)$, least $l$ making $2^n\!-\!1,\ldots,l^n\!-\!1$ pairwise coprime (Erdős #770)

For a positive integer $n$, let $h(n)$ be the least integer $l$ such that $2^n-1,\,3^n-1,\,\ldots,\,l^n-1$ are mutually coprime (the listed values are pairwise coprime). Erdős asked three linked questions: (i) does the density $\delta_p$ of the set of integers $n$ with $h(n)=p$ exist for every prime $p$; (ii) is $\liminf_{n\to\infty}h(n)=\infty$; and (iii) is it true that if $p$ is the greatest prime with $p-1\mid n$ and $p>n^{\epsilon}$, then $h(n)=p$?

Acceptance. FULLY RESOLVES: complete proofs (Lean/Coq preferred, else full written) settling the stated questions — establishing existence or non-existence of each density $\delta_p$, deciding whether $\liminf h(n)=\infty$, and proving or refuting the characterisation ('$p$ greatest prime with $p-1\mid n$ and $p>n^{\epsilon}$ $\Rightarrow h(n)=p$'). A definitive proof of any one of the three named sub-questions resolves that sub-question. ADVANCES: prove $h(n)=3$ for infinitely many $n$ (equivalently $\gcd(2^n-1,3^n-1)=1$ infinitely often); prove a nontrivial lower bound on the density of $\{n:h(n)=p\}$ for some fixed prime $p$; or a reproducible computation extending the tabulated values of $h(n)$ and of $\{n:h(n)=3\}$ (OEIS A263647) to a new certified range, with the search code and the observed density statistics. Deliver the proof(s), or the computation code plus certified extended tables and density estimates.