The asymptotic Turán number conjecture for the triangular pyramid of 4 layers

Let $TP_4$ denote the triangular pyramid of $4$ layers, and let $\operatorname{ex}(n,TP_4)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $TP_4$. Asymptotic Turán number conjecture for $TP_4$. For $n$ sufficiently large,

$$\operatorname{ex}(n,TP_4)=\frac{n^2}{4}+\Theta(n^{4/3}).$$

This conjecture concerns the asymptotic order of the excess over the balanced bipartite bound, and its resolution is not supplied in the source.