Exponential graph-size conjecture for alternating-group quotients

Let $X$ be a graph with $n$ vertices, and let $\operatorname{Aut}(X)$ be its automorphism group. Let $A_k$ denote the alternating group on $k$ letters.

Exponential graph-size conjecture. There exists a constant $C>1$ such that, if $k\geq3$ and $\operatorname{Aut}(X)$ has an epimorphism onto $A_k$, then

$$n\geq C^k.$$

The conjecture is motivated by Liebeck's result that $n\geq2^k-k-2$ for $k\geq13$ when $\operatorname{Aut}(X)\cong A_k$, with this bound tight for $k\equiv0$ or $1\pmod4$. The broader quotient formulation is left as a conjecture in the supplied text.