Conlon–Lee conjecture on improved extremal bounds for bipartite graphs

Let $H$ be a $K_{r,r}$-free bipartite graph with degree at most $r$ on one side.

Conlon–Lee conjecture. There exists $c=c_H>0$ such that

$$\operatorname{ex}(n,H)=O(n^{2-1/r-c}).$$

This strengthens the classical bound $\operatorname{ex}(n,H)=O(n^{2-1/r})$ for bipartite graphs with degree at most $r$ on one side. The conjecture is confirmed for hypercubes and bipartite Kneser graphs, but remains open in general.