The Turán number conjecture for the triangular pyramid of 3 layers

Let $TP_3$ denote the triangular pyramid of $3$ layers, and let $\operatorname{ex}(n,TP_3)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $TP_3$. Turán number conjecture for $TP_3$.

$$\operatorname{ex}(n,TP_3)\leq \begin{cases} \frac{1}{4}n^2+n+1, &\text{if $n$ is even,}\\ \frac{1}{4}n^2+n+\frac{3}{4},&\text{ otherwise}. \end{cases}$$

The conjecture is motivated by two constructions that provide lower bounds; determining the exact Turán number remains open.