Inverse restricted Erdős–Heilbronn conjecture below the prime threshold

Let $G$ be an abelian group, let $p$ be the smallest prime divisor of the order of $G$, and let $A$ be an $m$-subset of $G$. Assume that $m$ and $h$ are positive integers satisfying $p>hm-h^2+1$. Inverse restricted Erdős–Heilbronn conjecture. If $hA$ has size $hm-h^2+1$, then and only then either $h=1$, $A$ is an arithmetic progression, or $h=2$, $m=4$, and

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

for some $a\in A$ and $g_1,g_2\in G$. This is the restricted-addition analogue of the stated inverse results for unrestricted sumsets and remains open.