The asymptotic Turán conjecture for the square of a path

Let $P_k$ be the path on $k$ vertices, let $P_k^2$ denote its square, and let ${\rm ex}(n,P_k^2)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $P_k^2$. Here $o(1)$ denotes a quantity depending only on $k$ that tends to zero as $n$ tends to infinity. The asymptotic Turán conjecture.

$${\rm ex}(n,P_k^2)=\frac{n^2}{4}+\frac{\left(\left\lfloor {2k\over 3}\right\rfloor-2\right)n}{2}+o(1).$$

This is explicitly presented as a weaker form of the preceding conjecture, and is supported in the paper by the exact cases $k=3,4,5,6$.