Planarity conjecture for quasi-transitive graphs quasi-isometric to minor-excluded graphs

A graph $\Gamma$ is minor-excluded if some finite graph is not a minor of $\Gamma$. Two graphs are quasi-isometric when their large-scale metric spaces are equivalent up to multiplicative and additive distortion. Planarity conjecture for minor-excluded quasi-isometry classes. Let $\Gamma$ be a connected, locally finite, quasi-transitive graph, and suppose $\Gamma$ is quasi-isometric to a minor-excluded graph. Then $\Gamma$ is quasi-isometric to a planar graph. The quasi-transitivity assumption is necessary, and this conjecture would imply the preceding accessibility conjecture; its verification for one-ended graphs would make the two conjectures equivalent.