Almost-all non-congruence presentations conjecture for finite simple groups

Let $S$ be a non-abelian finite simple group. A presentation of $S$ is an epimorphism $F_2\rightarrow S$, and a presentation is non-congruence if its kernel does not arise from a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$.

Almost-all non-congruence presentations conjecture. As the order of $S$ tends to infinity, the proportion of non-congruence presentations among all presentations tends to $1$:

$$\lim_{|S|\rightarrow\infty}\frac{\#\text{non-congruence presentations}}{\#\text{all presentations}}=1.$$

The conjecture formalizes the expectation that congruence presentations are rare, despite the existence of infinite families of finite simple groups with characteristic congruence presentations. A congruence presentation would yield a functorial construction of a non-abelian finite simple cover of an elliptic curve from an isogeny, ramified only above the origin; the conjecture asserts that such presentations become negligible among all presentations as the group order grows.