Ghosh–Győri–Paulos–Xiao–Zamora's extremal conjecture for the triangular pyramid $TP_4$

Let $TP_4$ denote the triangular pyramid graph with four levels, and let $\operatorname{ex}(n,TP_4)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $TP_4$. Ghosh–Győri–Paulos–Xiao–Zamora's conjecture. For $n$ sufficiently large,

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

This conjecture concerns the asymptotic Turán number of the triangular pyramid graph $TP_4$; the exact extremal structure and the asserted asymptotic estimate remain unresolved in the source.