The asymptotic uniform distribution of subset sums

Let $G$ be a finite abelian group of order $n$, and for each $a\in G$ and integer $1\le h\le n$ let ${\mathcal{F}}_a(h)$ denote the family of all $h$-element subsets of $G$ whose sum is $a$. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families ${\mathcal{F}}_a(h)$ (as $a$ ranges over $G$) become asymptotically equal as $n\to \mathrm{\infty }$ when $h=\lfloor \frac{n}{2}\rfloor$. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every $4\le h\le \lfloor \frac{n}{2}\rfloor +1$.