Gao–Han–Peng–Sun conjecture for Erdős–Ginzburg–Ziv constants

Let $G$ be a finite abelian group, let $\beta=\exp(G)$ denote its exponent, and let $k\in\mathbb N$. The invariant $\mathsf{s}_{k\beta}(G)$ is the smallest length guaranteeing a zero-sum subsequence of length $k\beta$, while $\mathsf D(G)$ is the Davenport constant. Gao–Han–Peng–Sun conjecture. If $k\beta\ge \mathsf D(G)$, then

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

The conjecture extends Gao’s result from the condition $k\exp(G)\ge |G|$ to the weaker condition $k\exp(G)\ge\mathsf D(G)$. The source states that it remains open in specific rank-three $2$- and $3$-group cases.