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 $\mathsf{s}_{\leq k}(G)$ the smallest positive integer $l\in \mathbb{N}\cup \{+\infty\}$ 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 $\mathsf{s}_{\leq \mathsf{D}(G)-d}(G)\leq \mathsf{D}(G)+d$ for $\mathsf{D}(G)-d\geq k_G$. We conjecture that $k_G=\frac{\mathsf{D}(G)+1}{2}$ for finite abelian groups $G$ with $r(G)\geq 2$ and $\mathsf{D}(G)=\mathsf{D}^*(G)$. 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 $k_G\leq \mathsf{D}(G)-2$ for all finite abelian groups $G$ with $r(G)\geq 2$ except $C_2^3$ and $C_2^4$. In addition, we also get some lower bounds for the invariant $\mathsf{s}_{\leq k}(G)$.