Alon–Krivelevich–Sudakov conjecture for K_{s,t}-free graphs

Let $G$ be a graph with $m$ edges, and let $\mathrm{sp}(G)=\mathrm{mc}(G)-m/2$ denote its surplus, where $\mathrm{mc}(G)$ is the number of edges in a largest bipartite subgraph of $G$. For integers $t\ge s\ge2$, let $K_{s,t}$ be the complete bipartite graph with parts of sizes $s$ and $t$, and let $\mathrm{sp}(m,K_{s,t})$ be the smallest surplus among all $K_{s,t}$-free graphs with $m$ edges. Alon–Krivelevich–Sudakov conjecture. For all $t\ge s\ge2$ and all $m$,

$$\mathrm{sp}(m,K_{s,t})=\Omega\left(m^{\frac34+\frac{1}{8s-4}}\right).$$

This is a special case of the general surplus conjecture for forbidden graphs. The source presents it as open and discusses bounds for this family.