Face-extension conjecture for non-commutative crepant resolutions

Let $Q\subset\mathbb{R}^k$ be a lattice polytope with associated Gorenstein cone

$$\sigma=\operatorname{Cone}(Q\times\{1\})\subset\mathbb{R}^{k+1}.$$

Suppose $Q$ is lattice equivalent to a face $F$ of a lattice polytope $P\subset\mathbb{R}^n$. Let

$$\sigma'=\operatorname{Cone}(P\times\{1\})\subset\mathbb{R}^{n+1}$$

and let $M'$ be the $(n+1)$-dimensional character lattice of $X_{\sigma'}$. If $R_{\sigma'}=k[(\sigma')^\vee\cap M']$ has an NCCR, then $R_\sigma=k[\sigma\cap M]$ has an NCCR. Face-extension conjecture. Under these hypotheses, $R_\sigma$ admits an NCCR. The claim would reduce the existence problem for general Gorenstein cones to suitable reflexive or higher-dimensional ambient cones; the source presents it as an expectation and gives no resolution.