Generalized Turán number exponent conjecture

Let $s,a,b,t$ be integers satisfying $s<a\leq b$. For graphs $K_{a,b}$ and $K_{s,t}$, write $\mathrm{ex}(n,K_{a,b},K_{s,t})$ for the maximum number of copies of $K_{a,b}$ in an $n$-vertex $K_{s,t}$-free graph.

Generalized Turán number exponent conjecture. There exists $t_0$ such that, for all $t\geq t_0$,

$$\mathrm{ex}(n,K_{a,b},K_{s,t})=\Theta_{s,t,a,b}(n^s).$$

The preceding theorem proves this assertion for $s\in\{2,3\}$ when $t$ is sufficiently large. The conjecture extends that result to all $s<a\leq b$, and the source gives no resolution in the general case.