Coarse separability characterization for graph products of virtually nilpotent groups

Let $G=(V,E)$ be a finite graph, and let the groups $\Gamma_u$, $u\in V$, be infinite finitely generated virtually nilpotent groups. The graph product is

$$\Gamma(G)=\left(\mathop{\ast}_{u\in V}\Gamma_u\right)\big/\langle\langle [\Gamma_u,\Gamma_v]:\{u,v\}\in E\rangle\rangle.$$

A clique is a complete subgraph of $G$, and a disconnecting clique is a clique whose removal disconnects $G$.

Coarse separability conjecture. The graph $G$ has a disconnecting clique if and only if the graph product $\Gamma(G)$ is coarsely separable by a family of subexponential growth.

This is presented as a conjectural characterization in the setting of graph products with virtually nilpotent vertex groups; the source gives no resolution status or additional evidence for the claim.