Restricted sumset formula for the small-set regime

Let $G$ be a finite abelian group of order $n$, let $p$ be the smallest prime divisor of $n$, and let $\rho\hat{\;} (G,m,h)$ denote the minimum restricted $h$-fold sumset size among $m$-subsets of $G$. Small-set restricted-sumset conjecture. If $h<m\leq p$, then

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

This extends the prime-order restricted sumset formula of Dias da Silva and Hamidoune to the stated small-set regime in arbitrary finite abelian groups; the source presents it as conjectural.