Non-commutative crepant resolutions for (almost) simplicial toric algebras

Given a rational convex polyhedral Gorenstein cone constructed as cone over a lattice polytope P, we establish that toric non-commutative crepant resolutions (NCCRs) of its associated toric algebra descend to toric NCCRs of the algebras associated to faces of the polytope P. As consequence, we present two new, short proofs to the existence of toric NCCRs for simplicial affine toric Gorenstein algebras and for almost simplicial affine toric Gorenstein algebras, i.e. those associated to cones $\sigma$ with $\dim\sigma+1$ extremal rays.