The strong Seymour vertex conjecture for oriented graphs

Let $D$ be an oriented graph. For a vertex $x$, let $N_D^+(x)$ and $N_D^{++}(x)$ denote its out-neighborhood and second out-neighborhood, respectively. A complete matching from $X$ to $Y$ is a matching consisting of arcs directed from vertices of $X$ to vertices of $Y$ and having every vertex of $X$ as a tail. A vertex $x$ is a strong Seymour vertex if there is a complete matching from $N_D^+(x)$ to $N_D^{++}(x)$. Strong Seymour vertex conjecture. Every oriented graph contains a strong Seymour vertex. This would strengthen Seymour's second neighborhood conjecture, since a complete matching injects $N_D^+(x)$ into $N_D^{++}(x)$. It is proved in the paper for oriented graphs with minimum out-degree at most $5$ and for $5$-anti-transitive oriented graphs, and hence for oriented planar graphs; the general case remains open.