The small-set 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 $A^g=g^{-1}Ag$. Small-set asymmetric growth conjecture. For every $\epsilon>0$ there exists $\delta>0$ such that, whenever $|B|,|A|\le |S|^\delta$, there is some $g\in S$ for which

$$|BA^g|\ge |B|\cdot |A|^{1-\epsilon}.$$

The excerpt presents this as a proposed strengthening for smaller subsets; it supplies no resolution status or further known cases.