Inverse restricted-sumset conjecture above the small-set threshold

Let $G$ be a finite abelian group of order $n$, let $p$ be the smallest prime divisor of $n$, and let $A$ be an $m$-subset of $G$. Inverse restricted-sumset conjecture. If $2\leq h\leq m-2$ and $p>hm-h^2+1$, then

$$|h\hat{\;}A|=hm-h^2+1$$

if and only if either $h=2$, $m=4$, and

$$A=\{a,a+g_1,a+g_2,a+g_1+g_2\}$$

for some $a,g_1,g_2\in G$, or $A$ is an arithmetic progression in $G$. This is the proposed inverse classification corresponding to the restricted lower-bound formula; it remains open in the generality stated.