The anti-directed cycle Turán conjecture

Let $\overrightarrow{C_{2k}^a}$ be the anti-directed cycle on $2k$ vertices, let ${\rm ex}(n,\overrightarrow{C_{2k}^a})$ denote the maximum number of arcs in an $\overrightarrow{C_{2k}^a}$-free digraph of order $n$, and let ${\rm EX}(n,\overrightarrow{C_{2k}^a})$ be the family of extremal digraphs. For a digraph $D$, let ${\rm Brp}(D)$ be its associated bipartite graph, and let ${\rm ex}(n,n;C_{2k})$ and ${\rm EX}(n,n;C_{2k})$ denote the corresponding bipartite Turán number and family of extremal graphs.

The anti-directed cycle Turán conjecture. For every $k\in \mathbb{N}^*$, there exists a constant $n_0$ such that for $n\ge n_0$,

$${\rm ex}(n,\overrightarrow{C_{2k}^a})={\rm ex}(n,n;C_{2k}),$$

and

$${\rm EX}(n,\overrightarrow{C_{2k}^a})=\{D: {\rm Brp}(D)\in {\rm EX}(n,n;C_{2k})\}.$$

The case $k=2$ is already known through the equality with the Zarankiewicz number for $C_4$, while the asserted asymptotic equality and extremal-family correspondence for every $k$ remain open.