Füredi–Hajnal conjecture for arbitrary forbidden patterns

Let $P\in\{0,1\}^{k\times l}$ be any forbidden pattern, and let $G(P)$ be its unordered bipartite graph. Let $\operatorname{Ex}(P,n)$ denote the maximum number of ones in an $n\times n$ matrix avoiding $P$, and let $\operatorname{Ex}_{\mathit{Tur}}(G(P),n)$ denote the Turán extremal function of $G(P)$. Füredi–Hajnal logarithmic conjecture. One should have

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

Pach and Tardos refuted this conjecture in 2005, so it is no longer open.