Hall-type characterization of volume rigidity

Prove that, for a k-dimensional simplicial complex X on nd+2 vertices with |X_k|=dn-\binom{d+1}{2}, X is k-volume rigid in \mathbb{R}^d if and only if, for every subset S\subset X_k, the 1-skeleton of the restriction X[S] has rank at least |S| in the standard d-rigidity matroid.

Background

The second conjecture is presented as an equivalent Hall- or Rado-type reformulation of the first conjecture in the case where the number of k-simplices equals the maximum possible generic rank dn-\binom{d+1}{2}. In that setting, volume rigidity would be equivalent to the existence of a minimally d-rigid edge set that admits a perfect matching with the k-simplices through incidence.

The paper proves the only-if direction: for every collection S of k-simplices, the corresponding rows of the volume rigidity matrix factor through the rigidity matrix of the 1-skeleton of X[S], so their rank cannot exceed the standard rigidity-matroid rank of that 1-skeleton. The converse remains conjectural.