The global rigidity characterisation conjecture for frameworks on a circular cylinder

Let $G=(V,E)$ be a graph with $|V|\geq 5$, let ${\mathcal{M}}$ be a circular cylinder, and let $p$ be generic for ${\mathcal{M}}$. The global rigidity characterisation conjecture. The following are equivalent:

$$\begin{array}{ll} (1) & (G,p)\text{ is globally rigid on }{\mathcal{M}},\\ (2) & G\text{ is 2-connected and }(G,p)\text{ is redundantly rigid on }{\mathcal{M}},\\ (3) & G\text{ can be formed from disjoint copies of }K_5\setminus e,\ K_4\sqcup K_4\text{ and }K_4\veebar K_4\text{ by Henneberg 2 moves, 1-sums, 2-sums, 3-sums and edge additions},\\ (4) & G\text{ is }{\mathcal{R}}_{\mathcal{M}}\text{-connected}. \end{array}$$

The equivalence of (2) and (4) is established in the source, while the remaining combinatorial implication involving the listed construction moves is identified as the unresolved difficulty. The conjecture is an analogue of the known characterisation of generic global rigidity in the plane.