Coarse Thomassen theorem for vertex-transitive graphs

Let $G$ be a locally finite, vertex-transitive graph. A graph is quasi-isometric to a planar graph if its graph metric is quasi-isometric to the metric of some planar graph, and an asymptotic minor is a coarse minor model.

Coarse Thomassen theorem. Either $G$ is quasi-isometric to a planar graph, or $G$ contains every finite graph as an asymptotic minor.

This is proposed as a coarse version of Thomassen's result that a one-ended vertex-transitive graph is either planar or has an infinite clique as a minor. The source gives no resolution of the coarse assertion.