Akaria–Yuster's directed triangle packing conjecture for regular tournaments

Let $T$ be a regular tournament on $n$ vertices, where $n$ is odd. A collection of directed triangles is pairwise edge-disjoint if no directed edge belongs to more than one triangle. Akaria–Yuster's conjecture. Every regular tournament $T$ on $n$ vertices contains a collection of $n^2/9 + o(n^2)$ pairwise edge-disjoint directed triangles. The conjecture is motivated by the cyclic regular tournament, which attains the stated upper bound; the source notes that proving the bound for that tournament requires work and gives no resolution status.