Ailon–Rudnick conjecture on coprime values of exponential recurrences

Let $a,b$ be two multiplicatively independent non-zero integers satisfying

$$\gcd(a-1,b-1)=1.$$

Define the linear recurrences $F(n)=a^n-1$ and $G(n)=b^n-1$, and let

$$\mathcal{A}_{F,G}:=\{n\in\mathbb{N}:\gcd(F(n),G(n))=1\}.$$

Ailon–Rudnick conjecture. The set $\mathcal{A}_{F,G}$ is infinite.

This conjecture concerns the existence of infinitely many indices at which the values of two multiplicatively independent exponential recurrences are coprime. It is presented as an open problem; more general questions about the infinitude of sets defined by prescribed gcds of arbitrary integral linear recurrences are currently beyond the cited methods.