McDiarmid–Mohar conjecture on acyclic chromatic number of oriented graphs

Let $D$ be an oriented graph, meaning that at most one directed edge joins any pair of vertices. Write $\bar{\Delta}(D)$ for its average maximum degree and let $\chi_A(D)$ denote its acyclic chromatic number.

McDiarmid–Mohar conjecture. Every oriented graph $D$ satisfies

$$\chi_A(D) = O\left(\frac{\bar{\Delta}(D)}{\log \bar{\Delta}(D)}\right).$$

This conjecture predicts a logarithmic improvement over the greedy bound $\chi_A(D)\leq \bar{\Delta}(D)+1$, in analogy with strong upper bounds for chromatic number in undirected graphs. The source presents it as an open conjecture attributed to McDiarmid and Mohar.