Erdős's conjecture on the dichromatic number of digon-free digraphs

Let $D$ be a digon-free digraph, meaning that no pair of opposite directed edges joins the same two vertices, and let $\Delta$ be its maximum total degree. Erdős's conjecture. There is an absolute constant such that

$$\vec{\chi}(D)=O\left(\frac{\Delta}{\log_2 \Delta}\right).$$

This is the directed analogue of logarithmic improvements over the maximum-degree bound for chromatic number. The paper presents it as a conjecture of Erdős and studies a corresponding list-coloring upper bound; no resolution is supplied in the given text.