The set-of-length-sets conjecture for finite abelian groups

Let $G$ and $G'$ be finite abelian groups with $|G|,|G'|>3$, and let $\mathcal B(G)$ denote the block monoid over $G$. For a monoid $M$, write $\mathcal L(M)$ for its set of length sets.

Set-of-length-sets conjecture. If

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

then

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

This conjecture asks whether the set of length sets determines the block monoid, and hence the underlying finite abelian group, up to isomorphism. The source describes it as a famous conjecture and cites recent progress, but provides no resolution here.