The graph characterization of generic minimal rigidity on concentric cones and tori

Let ${\mathcal{M}}$ be a cone, a torus, a union of concentric cones or a union of concentric tori, and let $p$ be a generic placement. A graph $G$ is generically minimally rigid on ${\mathcal{M}}$ when the framework $(G,p)$ has generic minimal rigidity on ${\mathcal{M}}$. The graph $G$ is $(2,1)$-tight when it satisfies the corresponding $(2,1)$-tightness condition.

Graph characterization conjecture. Then $(G,p)$ is generically minimally rigid on ${\mathcal{M}}$ if and only if $G=K_{2}$, $K_3$, $K_4$, or $G$ is $(2,1)$-tight.

This conjecture would characterize generic minimal rigidity for frameworks on cones, tori, and unions of concentric cones or tori. The source presents it as further work expected to follow from the preceding inductive construction theorem; its resolution is not supplied here.