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 $F_a\left(h\right)$ 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 $F_a\left(h\right)$ (as $a$ ranges over $G$) become asymptotically equal as $n\to \infty$ when $h=\left(\frac{n}{2}\right)$. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every $4\le h\le \left(\frac{n}{2}\right)+1$.