Volume-rigidity conjecture for punctured sphere triangulations

Let $d\ge 3$, and let $K$ be a triangulation of the $(d-1)$-sphere. Removing a single $(d-1)$-simplex means deleting that facet and its interior. Higher-dimensional volume-rigidity conjecture. For every $d\ge 3$, every triangulation $K$ of the $(d-1)$-sphere minus a single $(d-1)$-simplex is volume rigid. For $2$-sphere triangulations, the corresponding exterior-shifting description is known and implies volume rigidity; this conjecture proposes the analogous statement in all higher dimensions.