The quasi-plane graph linear density conjecture

Call a topological graph $k$-quasi-plane if it has no $k$ pairwise crossing edges. The quasi-plane graph linear density conjecture. For any integer $k \geq 2$ there is a constant $c_k$ such that every $n$-vertex $k$-quasi-plane graph has at most $c_kn$ edges.

This conjecture would imply the linear edge bound for PCC simple topological graphs. It is known for $k=3$, for $k=4$, and for convex geometric graphs for every $k$, but remains open for $k \geq 5$.