Face invariance of Poincaré series for numerical semigroup algebras

Let $C_m$ be the Kunz cone, let $F$ be a face of $C_m$, and let $S$ and $T$ be numerical semigroups whose corresponding points lie in $F$. For a field $\Bbbk$, write $\Bbbk[S]$ and $\Bbbk[T]$ for the associated numerical semigroup algebras. Face-invariant Poincaré-series conjecture. The base field $\Bbbk$ has the same Poincaré series over $\Bbbk[S]$ and $\Bbbk[T]$ whenever $S$ and $T$ lie in the same face $F$ of $C_m$. This is presented as a final conjecture supported by computational evidence for small $m$, and would make the homological series an invariant of the Kunz face.