Inverse zero-sum conjecture for short zero-sum sequences in rank-two groups

Let $n\geq 2$, let $G=C_n \oplus C_n$, let $k\in [0,n-1]$, and let $S$ be a sequence of terms from $G$ with length $|S|=\operatorname{D}(G)+k-1=2n-2+k$ having no nonempty zero-sum subsequence of length at most $\operatorname{D}(G)-k=2n-1-k$. Then there exists a basis $(e_1,e_2)$ for $G$ such that the following hold. If $k=0$, then $S\boldsymbol{\cdot}g$ satisfies the description given in Item 2, where $g=-\sigma(S)$. If $k=1$, then

$$S=e_1^{[n-1]}\boldsymbol{\cdot}{\prod}^{\bullet}_{i\in[1,n]}{(x_ie_1+e_2)},$$

for some $x_1,\ldots,x_n \in [0,n-1]$ with $x_1+\cdots+x_n\equiv 1 \pmod{n}$. If $k\in [2,n-2]$, then

$$S=e_1^{[n-1]} \boldsymbol{\cdot} e_2^{[n-1]} \boldsymbol{\cdot} (e_1 + e_2)^{[k]}.$$

If $k=n-1$, then

$$e_1^{[n-1]} \boldsymbol{\cdot} e_2^{[n-1]} \boldsymbol{\cdot} (xe_1 + e_2)^{[k]},$$

for some $x\in[1,n-1]$ with $\gcd(x,n)=1$. This is the proposed structural characterization of all extremal sequences attaining the stated short zero-sum bounds; the cases $k\in\\{0,1,n-1\\}$ are known, while the intermediate range is the conjectural part.