Gain-tightness characterization for rotational and reflectional cylinder symmetries

Let $\mathcal{Y}$ denote the cylinder. Let $S$ be one of the groups $\mathcal{C}_{mh}$, $\mathcal{C}_{mv}$ or $\mathcal{S}_{2m}$, with the rotational axis equal to the cylinder axis. Let $(G,p)$ be an $S$-generic framework in the corresponding symmetric realization space $\mathscr{R}^{\mathcal{Y}}_{(G,S,\theta)}$, and let $(G_0,\psi)$ be its quotient $S$-gain graph. A gain graph is $(2,2,1)^r$-gain-tight when it is $(2,2,1)^r$-gain-sparse and has $|F|=2|V(F)|-1$ for its full edge set, with the bound $|F|\leq 2|V(F)|-2$ for nonempty $F$ whose vertex-generated subgroup is $\mathcal{C}_1$ or $\mathcal{C}_{m'}$ for $m'\leq m$, and $|F|\leq 2|V(F)|-1$ otherwise. The cylinder rotational-symmetry conjecture. The framework $(G,p)$ is $S$-isostatic if and only if $(G_0,\psi)$ is $(2,2,1)^r$-gain-tight. The cylinder summary records this as conjectural for the listed groups, so the characterization remains open in the source.