Volume rigidity of high-dimensional simplicial cones over 2-spheres

Prove that for every integer d >= 3, every simplicial 2-sphere S with ceil(d/2) + 2 vertices, and the simplicial cone S * Z formed by adjoining d - 3 new vertices, the (d-2)-skeleton hypergraph H_{d-2}(S * Z) is volume rigid in R^d.

Background

To extend their proofs for d = 4, 5, and 6 to all dimensions, the authors identify a base-case statement about cones over simplicial 2-spheres. The cone has d - 3 new vertices, and its relevant hypergraph is the (d-2)-skeleton hypergraph in Rd.

The vertex lower bound ceil(d/2) + 2 is necessary according to the paper’s edge-count argument: below that threshold, the number of volume constraints is too small for generic volume rigidity. Even proving this conjecture would not by itself complete the general codimension-one manifold conjecture, because an additional argument is needed for links with fewer vertices.

References

The inductive proof of these lemmas would extend to all $d$ if we could verify the following conjecture as the base case of our induction. \begin{conjecture} \label{lem:plane d-cone} Let $d\geq 3$ be an integer and $$ be a simplicial 2-sphere with $\lceil d/2\rceil+2$ vertices. Let $*Z$ be the simplicial cone of $$ with a set $Z$ of $d-3$ new vertices. Then $H_{d-2}(*Z)$ is volume rigid in ${d}$. \end{conjecture}

— Volume Rigidity of Simplicial Manifolds  (2503.01647 - Cruickshank et al., 3 Mar 2025) in Section 4, immediately before Section 5, Conjecture environment labeled Conjecture \ref{lem:plane d-cone}