The $k$-planar conjecture for quasi-transitive graphs

Let $\Gamma$ be a connected, locally finite, quasi-transitive graph. A graph is almost planar if it is $k$-planar for some $k\geq 0$, where $k$-planar means that it can be drawn in the plane so that each edge crosses at most $k$ other edges. Two graphs are quasi-isometric if they are quasi-isometric as metric spaces.

The $k$-planar conjecture. The graph $\Gamma$ is almost planar if and only if it is quasi-isometric to some planar graph.

This conjecture seeks a characterisation of almost planar, locally finite, quasi-transitive graphs in terms of quasi-isometry to planar graphs. It was posed by Georgakopoulos and Papasoglu and also appears in work of Esperet and Giocanti; its resolution is not established in the supplied source.