Fox–Pach sharp edge bound conjecture for K_{t,t}-free string graphs

For an integer $t\geq 2$, a graph is $K_{t,t}$-free if it contains no complete bipartite subgraph with $t$ vertices in each part. Fox–Pach's sharp edge bound conjecture. Every $K_{t,t}$-free string graph on $n$ vertices has at most

$$ct(\log t)n$$

edges, for an absolute constant $c$. The weaker linear bound is known with $c_t\leq t^{c\log\log t}$, while this $ct\log t$ estimate is conjectured to be best possible.