Cellular resolution conjecture for consistent toric algebras

Let $A$ be the consistent toric algebra associated to a collection on a Gorenstein affine toric variety $X$ of dimension $n$. Let $\mathbb{T}^n$ be the quotient torus in which the toric cell complex is constructed. Cellular resolution conjecture. If the global dimension of $A$ equals the dimension of $X$, then the toric cell complex $\Delta\subset\mathbb{T}^n$ exists and is constructed from the projected covering quiver as described in the paper. Moreover, the resulting complex is the minimal projective $(A,A)$-bimodule resolution of $A$. This conjecture extends the cellular-resolution construction from the examples treated in the paper to every consistent toric algebra whose global dimension equals the dimension of its underlying toric variety.