Rigid deformation conjecture for monomial ideals on regular CW-complexes

Let $I$ be a monomial ideal with a minimal resolution supported on a regular CW-complex having the intersection property, meaning that the intersection of any two cells is also a cell. Rigid deformation conjecture. Then $I$ admits a rigid deformation. This proposes that the stated intersection property is sufficient for the existence of a rigid deformation, extending the preceding result for the relevant poset constructions. The supplied text gives no resolution status or further evidence.