Zhao's conjecture on the short zero-sum threshold below the Davenport midpoint

Let $G$ be a finite abelian group with rank $r(G)\geq 2$ and $\mathsf{D}(G)=\mathsf{D}^*(G)$, and let $\exp(G)$ denote its exponent. Let $\mathsf{s}_{\leq m}(G)$ be the smallest length such that every sequence over $G$ has a nonempty zero-sum subsequence of length at most $m$. Zhao's conjecture. If $G\not\cong C_2^4$, $\mathsf{D}(G)-2\geq\exp(G)$, and $\exp(G)<(\mathsf{D}(G)-1)/2$, then

$$\mathsf{s}_{\leq \mathsf{D}(G)-2}(G)=\mathsf{D}(G)+1.$$

This refines the expected behavior of the bounded-length zero-sum invariant in a range below the midpoint of the Davenport constant.