The noncongruence-surjection conjecture for finite simple groups

Let $F_2$ be the free group of rank $2$, let $G$ be a nonabelian finite simple group, and let $\varphi:F_2\to G$ be a surjection. Denote by $\Gamma_\varphi$ the stabilizer of the corresponding point under the relevant $\operatorname{SL}_2(\mathbb{Z})$ action.

Noncongruence-surjection conjecture. If $G$ is a nonabelian finite simple group, then there exists a surjection $\varphi:F_2\to G$ such that $\Gamma_\varphi$ is noncongruence. Equivalently, the kernel of the $\operatorname{SL}_2(\mathbb{Z})$ action on $\operatorname{Epi}(F_2,G)/\operatorname{Inn}(G)$ is noncongruence.

This is proposed as a weakening of an earlier conjecture after the paper finds many noncongruence surjections and infinitely many counterexamples to that earlier statement. The proposed weakening remains open in the supplied text.