Parametrized minimal-resolution conjecture for Kunz faces

Let $C_m$ be the Kunz cone, let $F$ be a face of $C_m$, and let $S$ range over the numerical semigroups whose corresponding points lie in the interior of $F$. For each such $S$, let $I_S$ be the defining toric ideal. Parametrized minimal-resolution conjecture. For each face $F$ of $C_m$, there exists a parametrized family of minimal resolutions, one for $I_S$ for each numerical semigroup $S$ indexed by the interior of $F$, analogous to the families obtained for Apéry ideals. This would extend the known Apéry-resolution specialization phenomenon to defining toric ideals of numerical semigroups.