Fox–Pach separator conjecture for string graphs

A string graph is the intersection graph of a collection of curves in the plane. A separator in a graph $G=(V,E)$ is a subset $S\subseteq V$ such that no connected component of $G\setminus S$ has more than $\frac{2}{3}|V|$ vertices. Fox–Pach's separator conjecture. Every string graph with $m$ edges has a separator of order

$$O(\sqrt{m}).$$

This would remove the logarithmic factor from the previously known separator bound and would yield improvements to the paper's applications, including stronger edge bounds for $K_{t,t}$-free string graphs.