Generic and symmetry-forced isostaticity on the cylinder

Let $\mathcal{Y}$ denote the cylinder. Let $S$ be either the group $\mathcal{C}_s$ or the group $\mathcal{C}_i$, and let $(G,p)$ be an $S$-generic framework on $\mathcal{Y}$. The cylinder generic-equivalence conjecture. The framework $(G,p)$ is isostatic if and only if it is $S$-isostatic. The source motivates this by noting that the two representation blocks for these groups require the same $(2,2,1)$-gain-sparsity count; the proposed equivalence remains conjectural.