The characteristic-system-of-sets-of-lengths conjecture for finite abelian groups

Let $H$ be a transfer Krull monoid over a finite abelian group $G$, and let $\mathcal B(G)$ be the monoid of zero-sum sequences over $G$. Write $\mathsf L(a)$ for the set of factorization lengths of $a$ and

$$\mathcal L(H)=\{\mathsf L(a)\mid a\in H\},\qquad \mathcal L(G)=\mathcal L(\mathcal B(G)).$$

The system $\mathcal L(H)$ depends only on $G$. Characteristic-system-of-sets-of-lengths conjecture. If $G$ is a finite abelian group with $\mathsf D(G)\geq 4$ and $G'$ is an abelian group satisfying

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

then $G$ and $G'$ are isomorphic. This inverse problem asks whether the system of sets of lengths determines the class group; the conjecture is stated for the range $\mathsf D(G)\geq 4$, while the supplied text gives no resolution status.