Quasi-transitive $K_{\mathrm{\infty }}$-minor free graphs

We prove that every locally finite quasi-transitive graph that does not contain $K_{\mathrm{\infty }}$ as a minor is quasi-isometric to some planar quasi-transitive locally finite graph. This solves a problem of Esperet and Giocanti and improves their recent result that such graphs are quasi-isometric to some planar graph of bounded degree.