The $K_{4,6}$ counting conjecture for projective norm graphs

Let $\operatorname{NG}(q,4)$ be the projective norm graph over the parameter $q$, and let $K_{4,6}$ be the complete bipartite graph with parts of sizes $4$ and $6$. $K_{4,6}$ counting conjecture. The number of copies of $K_{4,6}$ in $\operatorname{NG}(q,4)$ is

$$\Theta(q^{16}).$$

This is motivated by the expectation that the count should have the same order as in a random graph with the same edge density; establishing the exact order remains open in the paper.