Recursive construction conjecture for redundantly rigid graphs on concentric cylinders

Let $G$ be a graph, and let $K_5-e$, $H_1$, and $H_2$ be the three specified base graphs. The operations under consideration are edge addition, $1$-extension, and $1$-, $2$- and $3$-join. Recursive construction conjecture. If $G$ is 2-connected and redundantly rigid on some, equivalently every, family of concentric cylinders, then $G$ can be obtained from one of $K_5-e$, $H_1$, or $H_2$ by recursively applying those operations. This conjecture would provide the graph-theoretic step toward proving sufficiency in the global-rigidity characterization for concentric cylinders.