Füredi–Hajnal conjecture for acyclic patterns

Let $P\in\{0,1\}^{k\times l}$ be an acyclic forbidden pattern, and let $G(P)$ be its unordered bipartite graph. Füredi–Hajnal acyclic-pattern conjecture. One should have

$$\operatorname{Ex}(P,n)=O\bigl(\operatorname{Ex}_{\mathit{Tur}}(G(P),n)\log n\bigr)=O(n\log n).$$

The paper presents this as a stronger predecessor of the Pach–Tardos conjecture and states that it was later refuted.