Volume conjecture for indecomposable summands of toric NCCRs

Let $R$ be the coordinate ring of an affine Gorenstein toric variety associated with a polytope $P$, and let a reflexive module define an NCCR of $R$. Denote by $\operatorname{Vol}(P)$ the volume appearing in the toric construction.

Toric summand-count conjecture. The number of indecomposable summands in the reflexive module defining an NCCR of $R$ is equal to

$$\operatorname{Vol}(P).$$

The conjecture is motivated by the rank of the Grothendieck group of the associated toric Deligne–Mumford stack. The source does not state a general proof or counterexample.