The extremal digraph conjecture for directed paths

Let $P_{k+1}$ be the directed path on $k+1$ vertices, let ${\rm ex}(n,\overrightarrow{P_{k+1}})$ denote the maximum number of arcs in an $\overrightarrow{P_{k+1}}$-free digraph of order $n$, and let ${\rm EX}(n,\overrightarrow{P_{k+1}})$ be the family of extremal digraphs. For $n\ge 3k$, let $\overrightarrow{T_{n,k}}$ be the corresponding $k$-partite transitive tournament construction, let $a(D)$ denote the number of arcs of a digraph $D$, and let $\overrightarrow{\mathcal{T}_{n,k}}$ be the family of its extremal constructions.

The extremal digraph conjecture for directed paths. For every $k,n\in \mathbb{N}^*$ with $n\ge 3k$,

$${\rm ex}(n,\overrightarrow{P_{k+1}})=a(\overrightarrow{T_{n,k}}),$$

and

$${\rm EX}(n,\overrightarrow{P_{k+1}})=\overrightarrow{\mathcal{T}_{n,k}}.$$

The result is known for directed paths of lengths corresponding to the established cases $k=2$ and $k=3$, but the general assertion for all $k$ and $n\ge 3k$ remains open.