Liebeck–Shalev conjecture on epimorphic homomorphisms from Fuchsian groups

Let $\Gamma$ be a Fuchsian group, and let $G$ be a finite simple classical group. A randomly chosen homomorphism in ${\rm Hom}(\Gamma,G)$ is an epimorphism with a probability that may depend on $G$. Liebeck–Shalev's conjecture. For any Fuchsian group $\Gamma$ there is an integer $f(\Gamma)$ such that, if $G$ has rank at least $f(\Gamma)$, then the probability that a randomly chosen homomorphism in ${\rm Hom}(\Gamma,G)$ is an epimorphism tends to $1$ as $|G|\rightarrow\infty$.

This extends asymptotic epimorphism results for Fuchsian groups and large finite simple groups; the source explicitly says that the conjecture remains open for arbitrary Fuchsian groups.