Structure conjecture for small quotient sets in groups

Let $A$ be a finite subset of a group $G$, and let $n$ be a positive integer. Write $N(H)$ for the normalizer of a subgroup $H$ in $G$. A subset $A_0$ is contained in a single left $N(H)$-coset if there is some $gin G$ such that $A_0\subseteq gN(H)$.

Structure conjecture. If

$$|A^{-1}A|<\left(2-\frac{1}{n+1}\right)|A|,$$

then there are a finite subgroup $H\le G$ and a subset $A_0\subseteq A$ of size $|A_0|\le n$ contained in a single left $N(H)$-coset such that

$$A\subseteq A_0H,\qquad |A_0H|=|A_0||H|,$$

and

$$|A|>\left(2-\frac{1}{n+1}\right)^{-1}(2|A_0|-1)|H|.$$

Moreover,

$$A^{-1}A=A_0^{-1}A_0H$$

and

$$|A^{-1}A|=(2|A_0|-1)|H|.$$

This conjecture proposes a structural description of finite subsets with quotient set smaller than the stated threshold. The supplied text gives no resolution status or further evidence beyond presenting it as a conjecture, so its status remains open.