Clique supersaturation conjecture for $K_{2,t}$

Let $K_{2,t}$ denote the complete bipartite graph with parts of sizes $2$ and $t$, and let $\mathcal{N}(F,G)$ denote the number of copies of $F$ in $G$. The $K_{2,t}$ clique supersaturation conjecture. There exists $t_0$ such that for $t\ge t_0$ and $1\le k\le n^{1/(2t)}$, there is an $n$-vertex graph $G$ with $\Omega(kn^{3/2})$ triangles and

$$\mathcal{N}(K_{2,t},G)\le k^t n^{3/2+o(1)}.$$

This predicts that the paper’s bounds for $K_{2,t}$ remain tight below the parameter range handled by its main construction.