Extension of the face theorem to non-toric NCCRs

Let $R$ be the toric algebra and let $\Lambda'$ be an NCCR of $R$, as in Theorem [the referenced theorem]. The theorem's statement should continue to hold when $\Lambda'$ is a non-toric NCCR.

Non-toric NCCR extension conjecture. The statement of the theorem holds also if $\Lambda'$ is a non-toric NCCR.

The preceding proof uses descent of rank $1$ reflexive modules from $R\otimes k[M_2]$ to $R$, and this argument does not apply in the same way to higher-rank modules. The conjecture asserts that the theorem nevertheless remains valid for NCCRs that are not toric.