Kawarabayashi–Picasarri-Arrieta dichromatic bound conjecture

Let $D$ be a digraph. Define

$$\widetilde{\Delta}(D)=\max_{v\in V(D)}\sqrt{d^-(v)d^+(v)},$$

and let $\overset{\text{tiny}}{\leftrightarrow}{\omega}(D)$ be the largest size of a biclique in $D$. Let $\vec{\chi}(D)$ denote the dichromatic number of $D$. Kawarabayashi–Picasarri-Arrieta conjecture. Every digraph $D$ satisfies

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

If true, this would imply Reed's conjecture and an independent conjecture of Harutyunyan and Mohar. The conjecture is presented as open in the source.