On zero-sum subsequences in a finite abelian group of length not exceeding a given number

Let $G$ be an additive finite abelian group and let $k\in [\exp (G),\mathsf{D}(G)-1]$ be a positive integer. Denote by ${\mathsf{s}}_{\le k}(G)$ the smallest positive integer $l\in \mathbb{N}\cup \{+\mathrm{\infty }\}$ such that each sequence of length $l$ over $G$ has a non-empty zero-sum subsequence of length at most $k$. Let $k_G\in [\exp (G),\mathsf{D}(G)-1]$ be the smallest positive integer such that ${\mathsf{s}}_{\le \mathsf{D}(G)-d}(G)\le \mathsf{D}(G)+d$ for $\mathsf{D}(G)-d\ge k_G$. We conjecture that $k_G=\frac{\mathsf{D}(G)+1}{2}$ for finite abelian groups $G$ with $r(G)\ge 2$ and $\mathsf{D}(G)={\mathsf{D}}^{\ast }(G)$. In this paper, we mainly study this conjecture for finite abelian $p$-groups and get some results to support this conjecture. We also prove that $k_G\le \mathsf{D}(G)-2$ for all finite abelian groups $G$ with $r(G)\ge 2$ except $C_2^3$ and $C_2^4$. In addition, we also get some lower bounds for the invariant ${\mathsf{s}}_{\le k}(G)$.