Volume rigidity of codimension-one faces of simplicial manifolds

Prove that for every d >= 3, every generic realization in R^d of the (d-2)-skeleton hypergraph of a connected simplicial (d-1)-manifold is volume rigid.

Background

The paper proves volume rigidity for the k-skeleton hypergraph of a connected simplicial (d-1)-manifold when 1 <= k <= d-3. The next case, k = d-2, concerns the hypergraph formed by the codimension-one faces of the manifold.

The authors explicitly conjecture that the same rigidity conclusion remains valid in this borderline case. They verify the conjecture for d = 3 using Fogelsanger’s result and establish it in the cases d = 4, 5, and 6, leaving the general-dimensional statement unresolved.

References

We conjecture that Theorem (iii) remains true when $k=d-2$. \begin{conjecture}\label{con:d-2} Let $H$ be the $(d-1)$-uniform hypergraph consisting of the $(d-2)$-faces of a connected simplicial $(d-1)$-manifold and $p$ be a generic realisation of $H$ in $d$ for some $d\geq 3$. Then $(H,p)$ is volume rigid. \end{conjecture}

— Volume Rigidity of Simplicial Manifolds  (2503.01647 - Cruickshank et al., 3 Mar 2025) in Introduction, Conjecture environment labeled Conjecture \ref{con:d-2}