Gluing characterization conjecture for globally linked pairs

Let $G=(V,E)$ be the union of graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ with $V_1\cap V_2=\{u,v\}$. 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 characterization conjecture. If $\{u,v\}$ is not linked in $G_2$ in $\mathbb{R}^d$, then $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$ if and only if it is globally linked in $G_1$ in $\mathbb{R}^d$.

The paper presents this as a counterpart to the planar gluing theorem and as a potential tool for recognizing $d$-entwined graphs algorithmically. The source gives no resolution, so the conjecture remains open.