The polylogarithmic conjecture for edge-ordered forests

An edge-ordered graph is a finite simple graph equipped with a linear order on its edges; its extremal function ${\mathrm{ex}}_<(n,H)$ is the maximum number of edges in an $n$-vertex edge-ordered graph avoiding $H$. Edge-ordered forest conjecture. For every edge-ordered forest $H$ of order chromatic number $2$,

$${\mathrm{ex}}_<(n,H)=n\log^{O(1)}n.$$

The paper proves the weaker bound ${\mathrm{ex}}_<(n,H)=n2^{O(\sqrt{\log n})}$, so improving it to a polylogarithmic factor is the main open problem identified by the authors.