The A5 composition-factor conjecture for solvabilizer number three

Let $G$ be a finite group, and let $\alpha(G)$ denote its solvabilizer number. A composition factor of $G$ means a factor in a composition series of $G$.

A5 composition-factor conjecture. If

$$\alpha(G)=3,$$

then $G$ has a composition factor isomorphic to $A_5$.

This conjecture is suggested by the computational data presented in the paper's appendix. It predicts that solvabilizer number three forces the presence of the alternating group $A_5$ among the composition factors.