Inverse restricted sumset conjecture at 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 $m\leq p<hm-h^2+1$. Inverse restricted sumset conjecture. The restricted $h$-fold sumset has size $p$ if and only if $A$ is contained in a coset of a subgroup $H\leq G$ with $|H|=p$. This is the second inverse problem paralleling the known unrestricted results and remains open.