Krause–Zahlten conjecture on the cross number of finite abelian groups

Let $G$ be a finite abelian group. For a sequence $S$ over $G$, let $\mathsf{k}(S)$ denote its cross number, let $\mathsf{k}(G)$ be the maximum cross number of a zero-sum free sequence over $G$, and let $\mathsf{K}(G)$ be the maximum cross number of a minimal zero-sum sequence over $G$. If $G=\bigoplus_{i=1}^{r}C_{p_i^{\alpha_i}}$ is written as a direct sum of prime-power-order cyclic groups, define

$$\mathsf{k}^{\star}(G)=\sum_{i=1}^{r}\left(1-\frac{1}{p_i^{\alpha_i}}\right),\qquad \mathsf{K}^{\star}(G)=\mathsf{k}^{\star}(G)+\frac{1}{\exp(G)}.$$

Krause–Zahlten conjecture. The equality

$$\mathsf{K}(G)=\mathsf{K}^{\star}(G)$$

holds for all finite abelian groups $G$, and therefore

$$\mathsf{k}(G)=\mathsf{k}^{\star}(G)$$

holds for all finite abelian groups $G$ as well. This is a central problem on extremal cross numbers of minimal zero-sum and zero-sum free sequences. The paper notes proofs for several families, including groups with restricted numbers of prime divisors in the exponent and groups of small rank, but states that a full proof remains far away.