Xiao–Katona–Xiao–Zamora conjecture on the Turán number of squared paths

Denote by $P_k$ the path on $k$ vertices. Its square, $P_k^2$, is obtained by joining every pair of vertices whose distance in $P_k$ is less than $2$. Let $\operatorname{ex}(n,H)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $H$.

Xiao–Katona–Xiao–Zamora conjecture. For the square of the path $P_k$, one has

$$\operatorname{ex}(n,P_k^2)\leq \max\left\{\frac{n_0\left(\left\lfloor\frac{2k}{3}\right\rfloor-2\right)}{2}+n_0n_1:n_0+n_1=n\right\}.$$

The paper states that this conjecture is settled, in a stronger form, by its characterization of the extremal graphs of powers of paths using a theorem of Simonovits.