Conjecture on squares of automorphism cosets in finite simple groups

Let $G=\operatorname{Inn}(G)$ be a non-abelian finite simple group, and let $\alpha\in\operatorname{Aut}(G)$. Coset-square conjecture. There exists $x\in\alpha G$ such that, for the conjugacy class $C=x^G$,

$$C^2=\alpha^2G.$$

This conjecture extends the idea that a suitable conjugacy class has maximal square expansion from inner classes to automorphism cosets. The source presents it as a conjecture supported by examples, without giving a resolution.