The large-r feedback arc set conjecture for r-free digraphs

Let $G=(V,E)$ be a digraph on $n$ vertices, let $\beta(G)$ denote the minimum size of a feedback arc set, and let an $(r-1)$-free digraph be one containing no directed cycle of length at most $r-1$. Large-r feedback arc set conjecture. If $G$ is an $(r-1)$-free digraph and $r>2n/3$, then

$$\beta(G)\leq 1.$$

The claim is presented as a more precise conjecture for the regime where $r$ is large relative to the number of vertices; no resolution is given.