Quasi-isometry classification conjecture for the groups W_Γ

Let $\Gamma$ and $\Gamma'$ be defining graphs, and let $W_\Gamma,W_{\Gamma'}\in\mathcal{W}$ be the associated groups.

Quasi-isometry classification conjecture. The groups $W_\Gamma$ and $W_{\Gamma'}$ are quasi-isometric if and only if they are isomorphic.

The preceding bounds distinguish some pairs of groups in $\mathcal{W}$ up to quasi-isometry, but do not give a complete classification. This conjecture proposes that isomorphism is exactly the quasi-isometry invariant for this family.