Schefler–Zhao–Zhong full-support conjecture for extremal separating atoms

Let $G=C_{n_1}\oplus\dots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\dots\mid n_r$. Let $A$ be a separating atom over a subset $G_0\subseteq G$ with $|G_0|\le r+1$, and suppose that $|A|=\beta_{\mathrm{sep}}(G)$.

Schefler56Zhao56Zhong's conjecture. The support of $A$ has full size:

$$|\operatorname{supp}(A)|=|G_0|=r+1.$$

The conjecture concerns the inverse problem for the separating Noether number of finite abelian groups: it predicts that every extremal separating atom uses all elements of the support permitted by Domokos' reduction. Exact formulas are known for several families, including groups of rank at most two and finite abelian $p$-groups, while a complete unconditional theory for arbitrary finite abelian groups remains open.