Short zero-sum sequences over abelian p-groups of large exponent

Let G be a finite abelian group with exponent n. Let η(G) denote the smallest integer ℓ such that every sequence over G of length at least ℓ has a zero-sum subsequence of length at most n. We determine the precise value of η(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.