Accessibility 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 $\Gamma$. A graph is accessible if it admits a finite decomposition over finite separators with pieces having at most one end. Accessibility 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 accessible. This is motivated by the known accessibility of connected, locally finite, quasi-transitive minor-excluded graphs and remains open under the stated quasi-isometry hypothesis.