Gluing conjecture for globally linked pairs

Let $d\geq 1$, and let $G$ be the union of graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. Suppose that $|V_1\cap V_2|\leq d+1$ and that $u,v\in V_1\cap V_2$. A pair is linked in a graph in $\mathbb{R}^d$ when its distance is fixed in every equivalent realization, and it is globally linked when this holds for every generic realization.

Gluing conjecture. If $\{u,v\}$ is linked in $G_i$ in $\mathbb{R}^d$ for $i\in\{1,2\}$, then $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$.

This conjecture extends the planar gluing theorem to higher dimensions and to gluings along vertex sets of size at most $d+1$. The source gives no resolution, so the conjecture remains open.