On some extensions of the Ailon-Rudnick Theorem

Let $a,b$ be multiplicatively independent positive integers and
$\varepsilon>0$.  Bugeaud, Corvaja and Zannier (2003) proved that
$$
\gcd (a^n-1,b^n-1)\le \exp(\varepsilon n)
$$
for a sufficiently large $n$. Moreover, Ailon and Rudnick conjectured
that when $\gcd(a-1,b-1)=1$, then
$\gcd (a^n-1,b^n-1)=1$  infinitely often. Using finiteness of the number
of torsion points on curves, Ailon and Rudnick (2004) proved the
function field analogue of this conjecture, in a stronger form, that is,
if $f,g\in\mathbb C[X]$ are multiplicatively independent polynomials, then
there exists
$h \in \mathbb C[X]$ such that for all $n\ge 1$ we have
$$
\gcd(f^n-1,g^n-1) \mid h.
$$