The depth-two Apéry-set conjecture for simplicial affine semigroup rings

Let $\mathcal{S}$ be the simplicial affine semigroup under consideration, let $\Bbbk[\mathcal{S}]$ be its semigroup ring, let $E$ be the distinguished generating set, and for $E'\subseteq E$ let $\operatorname{Ap}(\mathcal{S},E')$ denote the corresponding Apéry set, ordered by $\preceq_{\mathcal{S}}$. Depth-two Apéry-set conjecture. If

$$\operatorname{depth}(\Bbbk[\mathcal{S}])=2,$$

then there exists $E'\subseteq E$ with $|E'|=2$ such that $\operatorname{Ap}(\mathcal{S},E')$ has a maximal element with respect to $\preceq_{\mathcal{S}}$. This proposes a combinatorial characterization of depth two through maximal elements of Apéry sets; the supplied text presents it as an optimistic conjecture, with no resolution stated.