Ghosh et al.'s Turán-number conjecture for the triangular pyramid of 4-layers

Let $TP_4$ be the triangular pyramid with $4$ layers, and let $\textup{ex}(n,TP_4)$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $TP_4$. Ghosh et al.'s conjecture. For sufficiently large $n$,

$$\textup{ex}(n,TP_4)=\frac{1}{4}n^2+\Theta(n^{\frac{4}{3}}).$$

This conjecture predicts the second-order term in the Turán number of $TP_4$, extending the known asymptotic result for $TP_3$. Its resolution is not specified in the source.