Structural conjecture for extremal zero-sum sequences over finite abelian groups

Let $G$ be a finite abelian group, and let $p_1,p_2,\ldots,p_s$ be the primes dividing $\exp(G)$. For each $i$, call the subgroup consisting of elements whose orders are powers of $p_i$ the $p_i$-primary component of $G$. Let $\mathsf{k}(S)$ be the cross number of a sequence $S$, let $\mathsf{K}(G)$ be the maximum cross number of a minimal zero-sum sequence over $G$, and let $\mathsf{k}^{\star}(G)$ be the sum of $1-1/p^\alpha$ over the prime-power cyclic direct summands of $G$.

Structural conjecture. Each minimal zero-sum sequence $U$ over $G$ with $\mathsf{k}(U)=\mathsf{K}(G)$ has the form

$$U=g\prod_{i=1}^{s}U_i,$$

where $U_i$ is a zero-sum free sequence over the $p_i$-primary component of $G$ for each $i\in[1,s]$. In particular,

$$\mathsf{K}(G)=\frac{1}{\exp(G)}+\mathsf{k}^{\star}(G),$$

and every zero-sum free sequence $S$ with $\mathsf{k}(S)=\mathsf{k}^{\star}(G)$ has the form

$$S=\prod_{i=1}^{s}S_i,$$

where $S_i$ is a zero-sum free sequence over the $p_i$-primary component of $G$ for each $i\in[1,s]$. This strengthens the preceding cross-number conjecture by predicting the structure of all extremal sequences. The statement is motivated by structural results proved in the paper for several families, but remains open in general, especially when both the rank and the number of prime divisors of $\exp(G)$ are large.