  Let $G$ be a finite abelian group with exponent $n$. Let $\eta(G)$ denote the
smallest integer $\ell$ such that every sequence over $G$ of length at least
$\ell$ has a zero-sum subsequence of length at most $n$. We determine the
precise value of $\eta(G)$ when $G$ is a $p$-group whose Davenport constant is
at most $2n-1$. This confirms one of the equalities in a conjecture by Schmid
and Zhuang from 2010.
