Global rigidity conjecture for graphs containing triangulations of surfaces

Let $G$ be a graph with a triangulation $T$ of a surface $S$ as a spanning subgraph. A graph is globally rigid when its generic realization is uniquely determined, up to Euclidean isometry, by its edge lengths. Global rigidity conjecture for triangulated surfaces. The graph $G$ is globally rigid if and only if $G$ is $4$-connected and, when $S$ has genus zero, $G\neq T$. The conjecture extends the planar triangulation result to arbitrary surfaces. It appeared previously as a question and has been verified when $S$ is the sphere, projective plane, or torus; the general case remains open.