Noncongruence conjecture for nonabelian finite simple groups

Let $G$ be a nonabelian finite simple group. A finite group is noncongruence if it cannot occur as the monodromy group of a congruence cover in the relevant moduli problem.

Noncongruence conjecture. Every nonabelian finite simple group $G$ is noncongruence.

This conjecture refines the observed tendency for highly nonabelian groups to be uniformized by subgroups of low congruence degree. It was motivated by computations for finite simple groups of order at most $175560$, with only a small number of congruence components observed.