Characterization conjecture for finite abelian groups by sets of lengths

Let $G$ be a finite abelian group, and let $\mathsf D(G)$ denote its Davenport constant. Let $\mathcal L(G)$ denote the system of sets of lengths associated with $G$. For an abelian group $G'$, assume that

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

Characterization conjecture. If $\mathsf D(G)\geq 4$, then $G$ and $G'$ are isomorphic.

This conjecture asserts that the system of sets of lengths determines every finite abelian group whose Davenport constant is at least $4$, answering the Characterization Problem affirmatively in that range. The supplied text does not establish the conjecture in full, so its resolution remains open.