Zhao's midpoint conjecture for bounded-length zero-sum constants

Let $G$ be a finite abelian group with rank $r(G)\geq 2$ and $\mathsf{D}(G)=\mathsf{D}^*(G)$. Let $k$ be a positive integer with $\mathsf{D}(G)-k\in[\exp(G),\mathsf{D}(G)]$. Let $\mathsf{s}_{\leq m}(G)$ denote the smallest length such that every sequence over $G$ has a nonempty zero-sum subsequence of length at most $m$. Zhao's midpoint conjecture. If $\mathsf{D}(G)-k\geq(\mathsf{D}(G)+1)/2$, then

$$\mathsf{s}_{\leq \mathsf{D}(G)-k}(G)\leq\mathsf{D}(G)+k.$$

If $\mathsf{D}(G)-k<(\mathsf{D}(G)+1)/2$, then

$$\mathsf{s}_{\leq \mathsf{D}(G)-k}(G)>\mathsf{D}(G)+k.$$

The paper notes that this holds for $G=C_3^3$ and $C_5^3$, and that the first assertion is known at the endpoint $\mathsf{D}(G)-k=\exp(G)$ for finite abelian $p$-groups.