The length-set invariance conjecture for block monoids

Let $G$ and $G'$ be finite Abelian groups, and let $\mathcal B(G)$ and $\mathcal B(G')$ denote their block monoids of zero-sum sequences. For a monoid $S$, write $\mathcal L(S)$ for its set of length sets.

Length-set invariance conjecture. If $|G|,|G'|>3$ and

$$\mathcal L(\mathcal B(G))=\mathcal L(\mathcal B(G')),$$

then

$$\mathcal B(G)=\mathcal B(G').$$

This conjecture asserts that, apart from the stated small-order exception, the set of length sets completely determines the block monoid among finite Abelian groups. The supplied text gives no information about whether the conjecture has been resolved.