The sparsity conjecture for graph independence in non-Euclidean p_q^d

Let $G=(V,E)$ be a graph, let $d\geq 1$, and define $f_d(G)=d|V|-|E|$. The graph $G$ is $(d,d)$-sparse if $f_d(H)\geq d$ for every subgraph $H\subset G$, and $(d,d)$-tight if it is $(d,d)$-sparse and $f_d(G)=d$. A graph is independent in $\ell_q^d$ when its associated rigidity matrix has independent rows.

Sparsity conjecture. For $q\in(1,\infty)$ with $q\not=2$, and $d\geq 1$, a graph $G$ is independent in $\ell_q^d$ if and only if $G$ is $(d,d)$-sparse.

This conjecture would characterize independence, and hence the combinatorial basis for minimal rigidity, in non-Euclidean $\ell_q^d$ spaces. The paper presents results supporting the corresponding conjecture that minimal rigidity is equivalent to $(d,d)$-tightness, but no resolution is supplied here.