On the inverse problem of the $k$-th Davenport constants for groups of rank $2$

For a finite abelian group $G$ and a positive integer $k$, let ${\mathsf{D}}_k(G)$ denote the smallest integer $\mathrm{\ell }$ such that each sequence over $G$ of length at least $\mathrm{\ell }$ has $k$ disjoint nontrivial zero-sum subsequences. It is known that ${\mathsf{D}}_k(G)=n_1+k{n}_2-1$ if $G\cong C_{n_1}\oplus C_{n_2}$ is a rank $2$ group, where $1<n_1\text{t}{n}_2$. We investigate the associated inverse problem for rank $2$ groups, that is, characterizing the structure of zero-sum sequences of length ${\mathsf{D}}_k(G)$ that can not be partitioned into $k+1$ nontrivial zero-sum subsequences.