Gain-tightness characterization for symmetric frameworks on the cone

Let $\mathcal{C}$ denote the cone. Let $S$ be either the group $\mathcal{C}_2$ representing a two-fold rotation about an axis perpendicular to the $z$-axis, or the group $\mathcal{C}_s$ whose mirror plane contains the $z$-axis. Let $(G,p)$ be an $S$-generic framework in the symmetric realization space $\mathscr{R}^{\mathcal{C}}_{(G,S,\theta)}$, and let $(G_0,\psi)$ be its quotient $S$-gain graph. The cone gain-tightness conjecture. The framework $(G,p)$ is $S$-isostatic if and only if $(G_0,\psi)$ is $(2,1,0)$-gain-tight. The source introduces this as a suggested conjecture for the cone; it concerns the symmetry groups for which the symmetry can force a distinction from generic rigidity.