Quasi-transitive 1-planar coarse-planarity conjecture

A graph is quasi-transitive if its automorphism group has finitely many vertex orbits. A graph is 1-planar if it has a plane drawing in which every edge is involved in at most one crossing, and bounded degree means that vertex degrees are uniformly bounded. A graph is quasi-isometric to a planar graph if it is quasi-isometric to some planar graph.

1-planar coarse-planarity conjecture. Every quasi-transitive 1-planar graph of bounded degree is quasi-isometric to a planar graph.

The source observes that proving this case would suffice for the preceding bounded-local-crossing conjecture. No resolution is given.