Gerbner–Nagy–Patkós–Vizer quasi-complete bipartite extremal conjecture

Let $H$ be a fixed bipartite graph. For positive integers $n,m$, let $N_{\mathrm{bip}}(n,m,H)$ be the maximum number of unlabelled copies of $H$ over bipartite graphs $G$ with $|V(G)|=n$ and $e(G)=m$. Let $B_n^m$ be the quasi-complete bipartite graph obtained by taking the smallest integer $t$ such that $t(n-t)\ge m$, starting from $K_{t,n-t}$, and deleting $t(n-t)-m$ edges all incident to one vertex in the part of size $t$. Gerbner–Nagy–Patkós–Vizer conjecture. If $m=\omega(n)$ and $m\le n^2/4$, then

$$N_{\mathrm{bip}}(n,m,H)=(1+o(1))N(B_n^m,H).$$

The conjecture proposes the quasi-complete bipartite graph as the asymptotic extremizer for copies of every fixed bipartite graph in a bipartite host with prescribed order and size. It is false in the subquadratic range.