Disjoint crossing families conjecture for geometric graphs

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. For integers $k,l\geq 1$, a $(k,l)$-crossing family is a pair of edge subsets $E_1,E_2$ such that $|E_1|=k$, $|E_2|=l$, the edges in each subset are pairwise crossing, and every edge in $E_1$ is disjoint from every edge in $E_2$. Disjoint crossing families conjecture. Given fixed constants $k,l\geq 1$, there exists a constant $c_{k,l}$ such that any geometric graph on $n$ vertices with no $(k,l)$-crossing family has at most $c_{k,l}n$ edges. The conjecture is an extremal question about forbidden crossing and disjointness patterns in geometric graphs. The case $(k,l)=(2,1)$ is proved with an $O(n)$ bound, while the general case remains open; the analogous assertion for topological graphs is false.