The small MCM conjecture for tame bi-partite toric rings

Let $S$ be a regular local ring of characteristic $p>0$, let $\Gamma\subset\mathbb N^n$ be a full semigroup, and let $\varphi\colon\Gamma\to\mathbb N^d$ and $\chi\colon\Gamma\to S^*$ be respectively a homomorphism and an $S$-character. Let $R:=\mathcal R(S,\Gamma,\varphi,\chi)$ be the bi-partite toric ring defined by this data. Write $\operatorname{sat}_{S}(*1;\mathbf{F}_{q*}R)$ for the indicated $S$-saturation in the Frobenius transform, and suppose, in the additional situation described in the source, that $R$ has a unique $d$-dimensional prime $\mathfrak p$ and that $\varphi$ is tame.

Small MCM conjecture for tame bi-partite toric rings. For some sufficiently high power $q$ of $p$, $R$ admits the small MCM

$$\operatorname{sat}_{S}(*1;\mathbf{F}_{q*}R).$$

Moreover, under the additional hypotheses, $\mathfrak p$ annihilates this module and

$$\operatorname{sat}_{S}(*1;\mathbf{F}_{q*}R)\cong \mathbf{F}_{\!}^{\operatorname{int}}(R/\mathfrak p)=(R/\mathfrak p)^{\operatorname{pow}}.$$

The statement extends the proposed small MCM construction beyond the principal families considered earlier. The source notes that the tameness assumption is necessary in an example; the conjecture itself remains open.