The asymptotic coloring conjecture for oriented graphs with bounded cycle lengths

Let $D$ be an oriented graph with no directed cycle of length greater than $s$, and let $\vec{\chi}(D)$ denote its dichromatic number.

Asymptotic coloring conjecture. As $s\rightarrow\infty$,

$$\vec{\chi}(D)=O\left(\frac{s}{\log s}\right).$$

This conjecture predicts an asymptotic improvement over the bound $\vec{\chi}(D)\leq \lceil s/2\rceil$ for oriented graphs of circumference at most $s$. The supplied source indicates that this statement has been proved, so it is recorded as solved.