The d-dimensional body-bar orbit framework rigidity conjecture

Let $\langle H,m\rangle$ be a body-bar orbit framework on the fixed torus $\mathcal T_0^d$, and let $\mathcal G_{\mathcal C}(Y)$ denote the collection of connected components associated with an edge set $Y$. Body-bar orbit framework rigidity conjecture. $\langle H,m\rangle$ is a generically minimally rigid body-bar framework on $\mathcal T_0^d$ if and only if

$$|E(H)|={d+1\choose 2}|V(H)|-d$$

and, for every non-empty subset $Y\subset E(H)$,

$$|Y|\leq {d+1\choose 2}|V(Y)|-{d+1\choose 2}+\sum_{i=1}^{|\mathcal G_{\mathcal C}(Y)|}(d-i).$$

This conjectures that the three-dimensional characterization proved in the paper extends to body-bar orbit frameworks in arbitrary dimension $d$.