The strong Seymour vertex conjecture for tournaments

Let $T$ be a tournament, namely an oriented graph in which every pair of distinct vertices is joined by exactly one directed arc. For a vertex $x$, let $N_T^+(x)$ and $N_T^{++}(x)$ denote its out-neighborhood and second out-neighborhood, respectively. A vertex $x$ is a strong Seymour vertex if there is a complete matching from $N_T^+(x)$ to $N_T^{++}(x)$. Tournament strong Seymour vertex conjecture. Every tournament contains a strong Seymour vertex. This is the tournament case of the stronger conjecture for oriented graphs. The statement is false: replacing each independent set in the paper's counterexample by a transitive tournament produces a $36$-vertex tournament with the same property.