Clique supersaturation conjecture for $K_{s,t}$ when $r\le s$

Let $K_{s,t}$ be the complete bipartite graph with parts of sizes $s$ and $t$, let $K_r$ be the complete graph on $r$ vertices, and let $\mathcal{N}(F,G)$ denote the number of copies of $F$ in $G$. The $K_{s,t}$ clique supersaturation conjecture. If $2\le r\le s\le t$, then there exists a constant $k_0$ such that every $n$-vertex graph $G$ satisfying $\mathcal{N}(K_r,G)=k n^{r-\binom{r}{2}/s}$ and $k\ge k_0$ also satisfies

$$\mathcal{N}(K_{s,t},G)\ge k^{st/\binom{r}{2}} n^{s-o(1)}.$$

The conjecture asserts that, in the range $r\le s$, every graph with the prescribed number of $K_r$ copies contains at least as many $K_{s,t}$ copies as the corresponding random-graph heuristic predicts. The source notes that the case $r=2$ follows from a proposition, making $r=s=3$ the next open case.