The systems-of-sets-of-lengths characterization of finite abelian groups

Let $G$ and $G'$ be abelian groups, and let $\mathcal L(G)$ denote the system of sets of lengths associated to $G$. Suppose that $G$ is finite and has Davenport constant

$$\mathsf D(G) \ge 4.$$

Systems-of-sets-of-lengths conjecture. If $G'$ is an abelian group satisfying

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

then $G$ and $G'$ are isomorphic. Equivalently, apart from the trivial cases listed for groups of order at most two, the system of sets of lengths determines the finite abelian group. This conjecture concerns the extent to which arithmetic information encoded by zero-sum sequences determines the underlying group; it is stated as far open in general.