The F-integral Cohen–Macaulay conjecture for complete toric domains

Let $R$ be a complete toric domain of positive characteristic, and let $\mathbf{F}_{\!}^{\operatorname{int}}(R)$ be its ring of F-integral elements.

F-integral Cohen–Macaulay conjecture. The ring $\mathbf{F}_{\!}^{\operatorname{int}}(R)$ is Cohen–Macaulay, and hence is a small MCM algebra over $R$.

For affine toric domains, normality implies Cohen–Macaulayness by Hochster's theorem; the conjecture proposes the corresponding statement for complete local toric domains. It remains open.