Uniform equivariant quasi-isometry conjecture for geometrically rigid hyperbolic groups

Let $G$ be a geometrically rigid hyperbolic group. An abstract commensurator of $G$ is an isomorphism $\phi:H_1\to H_2$ between finite-index subgroups $H_1,H_2\leq G$. A map $\Phi:G\to G$ is $\phi$-equivariant if it intertwines the actions of $H_1$ and $H_2$ via $\phi$, and it is a $(C,C)$-quasi-isometry when its quasi-isometry constants are both $C$. Uniform equivariant quasi-isometry conjecture. There exists a constant $C=C(G)$ such that every abstract commensurator $\phi:H_1\to H_2$ admits a $\phi$-equivariant map $\Phi:G\to G$ which is a $(C,C)$-quasi-isometry. This conjecture seeks to generalize the uniform geometric control available for commensurators of hyperbolic manifolds to all geometrically rigid hyperbolic groups; the source provides no resolution, so the conjecture remains open.