Approximate orbit structure conjecture for group actions

Let $G$ be a group acting on a set $X$, let $A$ be a subset of $G$, and let $Y$ be a finite subset of $X$. Write $A(Y)$ for the image of $Y$ under $A$, and let $K\geq 1$ satisfy $|A(Y)|\leq K|Y|$. Approximate orbit structure conjecture. There is a constant $C>0$, a subset $B\subseteq G$, a subgroup $H\leq G$, and a finite subset $Z\subseteq X$ such that

$$|B|\ll K^C,$$

$A\subseteq BH$, and

$$|H(Z)\cap Y|\gg K^{-C}|Y|.$$

This conjecture asks whether small image growth forces $Y$ to contain a substantial portion of an orbit-like set for a subgroup generated, up to a bounded set, by $A$. It is a group-action analogue of approximate structure results in multiplicative combinatorics; the statement is intended to relax the exact orbit decomposition obtained when $|A(Y)|=|Y|$.