The one-discus global-rigidity conjecture for discus-and-hole graphs

Let $G^\dagger$ be a discus-and-hole graph with exactly one discus. A framework is generically globally rigid in $\mathbb R^3$ when generic realizations are uniquely determined up to Euclidean isometries by their edge lengths, and a graph is redundantly rigid in $\mathbb R^3$ when it remains rigid after the deletion of any one edge.

One-discus global-rigidity conjecture. The graph $G^\dagger$ is generically globally rigid in $\mathbb R^3$ if and only if it is $4$-connected and redundantly rigid in $\mathbb R^3$.

This is motivated by the known difficulty of global rigidity and by a conjecture of Connelly and Whiteley on vertex splitting. The corresponding vertex-splitting conjecture is still open, although it has been verified in special cases; the proposed characterization for discus-and-hole graphs is therefore open.