The asymmetric growth conjecture

Let $S$ be a non-abelian finite simple group, and let $A,B$ be subsets of $S$. For $g\in S$, write $A^g=g^{-1}Ag$. Asymmetric growth conjecture. For every $\delta>0$ there exists $\epsilon>0$ such that, whenever $|B|\le |S|^{1-\delta}$, there is some $g\in S$ for which

$$|BA^g|\ge |B|\cdot |A|^\epsilon.$$

This conjecture is known when $A$ is a normal subset and for finite simple groups of Lie type, but remains open for alternating groups.