The linked-pairs dimension-dropping conjecture

Let $G=(V,E)$ be a graph. A pair of vertices is linked in $G$ in $\mathbb{R}^k$ when its distance is fixed across all generic equivalent realizations, and globally linked when that distance is fixed across all equivalent realizations. The linked-pairs dimension-dropping conjecture. If $\{u,v\}$ is linked in $G$ in $\mathbb{R}^{d+1}$, then $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$. This would clarify the relationship between rigidity in consecutive dimensions and would imply the paper's dimension-dropping theorem and a strengthening of its minimally globally rigid graph theorem. The conjecture is proved in the cases $d=1,2$ but remains open in general.