Gain-tightness characterization for half-turn symmetric frameworks on the cylinder

Let $\mathcal{Y}$ denote the cylinder, and let $S$ be the cyclic group $\mathcal{C}_2$ representing a two-fold rotation about an axis perpendicular to the $z$-axis. Let $(G,p)$ be an $S$-generic realisation on $\mathcal{Y}$, and let $(G_0,\psi)$ be its quotient $S$-gain graph. The half-turn cylinder conjecture. The framework $(G,p)$ is $S$-isostatic if and only if $(G_0,\psi)$ is $(2,2,0)$-gain-tight. This is listed as conjectural in the cylinder summary; it would characterize symmetry-forced rigidity for the perpendicular half-turn symmetry not covered by the established cylinder results.