Crossing-pairs conjecture for topological graphs

A topological graph is a graph drawn in the plane with vertices represented by points and edges by curves connecting their endpoints; two edges cross when their curves intersect away from common endpoints. The crossing-pairs conjecture. In every topological graph with $n$ vertices and $m\geq 4n$ edges, there are two disjoint sets of edges, each of cardinality

$$\Omega\left(\frac{m^2}{n^2\log\frac{m}{n}}\right),$$

such that every edge in one set crosses every edge in the other. The claimed bound would be tight and is presented as a consequence of the sharper extremal conjecture.