Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-1,3^n-1)=1$ infinitely often?)

Estimate $H(n)$. Is it true that there exists some constant $c>0$ such that, for all $\epsilon>0$, \[H(n) > \exp(n^{(c-\epsilon)/\log\log n})\] for infinitely many $n$ and \[H(n) < \exp(n^{(c+\epsilon)/\log\log n})\] for all large enough $n$?

Does a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?