Zero-sum invariants on finite abelian groups with large exponent

Let $G$ be an additive finite abelian group with exponent $n$. Let $\mathsf{D}\left(G\right)$ be the Davenport constant of $G$, ${\mathsf{s}}_{kn}\left(G\right)$ the $k$th Erd?s–Ginzburg–Ziv constant of $G$, where $k$ is a positive integer. Recently, Gao, Han, Peng and Sun conjectured that ${\mathsf{s}}_{kn}\left(G\right)=kn+\mathsf{D}\left(G\right)?1$ holds if $k\ge ?\frac{\mathsf{D}\left(G\right)}{n}?$. Let $m,n$ be positive integers and $H$ an abelian $p$-group with $\mathsf{D}\left(H\right)\le p^n$. Let $G=H\oplus C_{m{p}^n}$. For any integer $k\ge 2$, we prove that ${\mathsf{s}}_{km{p}^n}\left(G\right)=(k+1)m{p}^n+\mathsf{D}\left(H\right)?2=km{p}^n+\mathsf{D}\left(G\right)?1.$ This verifies the above conjecture in this case. We also provide asymptotically tight bounds for zero-sum invariants $\mathsf{D}\left(G\right)$, ${\mathsf{s}}_{kn}\left(G\right)$ and $\eta \left(G\right)$ for a class of abelian groups with large exponent.