Harutyunyan–Mohar conjecture for oriented graphs

Let $D$ be an oriented graph, meaning a digraph with no biclique of order greater than $1$. Let $\widetilde{\Delta}(D)=\max_{v\in V(D)}\sqrt{d^+(v)d^-(v)}$ be its maximum geometric mean of in- and out-degrees, and let $\vec\chi(D)$ be its dichromatic number, the minimum number of colours whose colour classes induce acyclic subdigraphs.

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

$$\vec\chi(D)\leq\left\lceil\frac{\widetilde{\Delta}(D)}{2}\right\rceil+1.$$

This is the proposed oriented-graph analogue of Reed's conjecture. The paper records partial upper bounds but leaves this conjecture open.