The global linkage characterization conjecture for rigid graphs

Let $G=(V,E)$ be a rigid graph and let $u,v\in V$. A pair of vertices is globally linked if every generic realisation of $G$ determines the distance between its two vertices. Global linkage characterization conjecture. The pair $\\{u,v\\}$ is globally linked in $G$ if and only if either $uv\in E$ or $u$ and $v$ are joined by three internally disjoint paths in some ${\mathcal{RM}}$-connected component of $G$. This would characterize when two vertices in a rigid graph are globally linked; the corresponding result is known when the whole graph is ${\mathcal{RM}}$-connected, but the general rigid-graph case remains open.