Face-dependent artinian reduction conjecture for semigroup resolutions

Let $C_m$ be the Kunz cone, let $F$ be a face of $C_m$, and let $S$ be a numerical semigroup whose corresponding point lies in $F$. Let $J_S$ be the ideal in the Apéry-resolution construction, and let $I_S$ be the defining toric ideal. Artinian-reduction conjecture. Substituting $y=0$ into the minimal resolution for $J_S$ obtained in the cited theorem, or analogously into the minimal resolution for $I_S$ predicted by the parametrized minimal-resolution conjecture, yields a minimal free resolution of an artinian binomial ideal depending only on the face $F$ containing $S$. This predicts that the specialization preserves minimality and removes dependence on the particular semigroup within the face.