Sullivan's conjecture on feedback arc sets in m-free digraphs

Let $m\ge 3$, and let $G$ be an $m$-free digraph, meaning that $G$ has no directed cycle of length at most $m$. Let $\beta(G)$ be the minimum number of edges whose removal makes $G$ acyclic, and let $\gamma(G)$ be the number of unordered pairs of nonadjacent vertices of $G$. Sullivan's conjecture.

$$\beta(G)\le \frac{2}{(m+1)(m-2)}\gamma(G).$$

Sullivan proposed this as a best-possible general bound. The paper states related results for several values of $m$, but does not report a resolution of the general conjecture.