The shared-stress-kernel witness conjecture

Let $G$ be a graph and let $u,v\in V(G)$ be a pair of vertices. Let $K(G,p)$ denote the shared stress kernel associated with a generic realization $(G,p)$. Suppose that $\{u,v\}$ is not globally linked in $G$ in $\mathbb{R}^d$. Shared-stress-kernel witness conjecture. For every generic realization $(G,p)$ in $\mathbb{R}^d$, there exist real configurations $q,q'\in K(G,p)$ such that $(G,q)$ is generic, $(G,q)$ and $(G,q')$ are equivalent, but $(G+uv,q)$ and $(G+uv,q')$ are not equivalent. This is proposed as a possible route toward detecting non-global linkage and is left open.