The acyclic 3-dicolouring conjecture for planar oriented graphs

An oriented planar graph is an orientation of a planar graph without a pair of opposite arcs. For an oriented graph $D$, let $\vec{\chi}_{\rm a}(D)$ denote the least number of colours in a vertex-colouring whose monochromatic subdigraphs are acyclic.

Planar acyclic dichromatic conjecture. Every oriented planar graph $D$ satisfies

$$\vec{\chi}_{\rm a}(D)\leq3.$$

The paper proves the upper bound $5$ and gives a planar oriented graph with acyclic dichromatic number at least $3$. Thus the conjectured bound would be tight if true, but its validity remains open.