Volume-rigidity conjecture for punctured surface triangulations

A triangulation of a compact connected surface without boundary is a simplicial complex representing the surface; removing a single triangle means deleting that triangular face and its interior. Volume-rigidity conjecture. Every triangulation of a compact connected surface without boundary, minus a single triangle, is volume-rigid. This would extend the known result for the currently covered class of surface triangulations; the obstruction is that Fogelsanger's decomposition introduces triangle faces absent from the original triangulation, beyond the strength of the available gluing lemmas.