Sufficient conditions for generic global rigidity on the cylinder and cone

Let $(G,p)$ be a generic framework on ${\mathcal{M}}$, where ${\mathcal{M}}\in\{{\mathcal{Y}},{\mathcal{C}}\}$. A framework is globally rigid on ${\mathcal{M}}$ when every framework on ${\mathcal{M}}$ equivalent to $(G,p)$ is congruent to it. Global rigidity conjecture for the cylinder and cone. The framework $(G,p)$ is globally rigid on ${\mathcal{M}}$ if and only if either $G$ is a complete graph on at most four vertices, or $G$ is $2$-connected and $(G,p)$ is redundantly rigid. The corresponding conditions are known to be sufficient for generic global rigidity in the plane and on the sphere; the conjecture asserts sufficiency in the cylinder and cone cases as well.