Restricted Erdős–Heilbronn formula for abelian groups

Let $G$ be an abelian group, and let $p$ be the smallest prime divisor of the order of $G$. For positive integers $m$ and $h$ with $m\leq p$, write $\rho\hat{\;} (G,m,h)$ for the minimum size of a restricted $h$-fold sumset of an $m$-element subset. Restricted Erdős–Heilbronn conjecture.

$$\rho \hat{\;} (G,m,h)=\min\{p,hm-h^2+1\}.$$

The formula is known for groups of prime order, while the corresponding statement for general abelian groups is presented as a potential next case and remains open.