Polynomial-time acyclic k-dicolouring conjecture for tournaments

For a tournament $T$, let $\vec{\chi}_{\rm a}(T)$ denote its acyclic dichromatic number. A fixed integer $k\geq3$ is treated as a constant rather than part of the input.

Tournament acyclic dicolouring algorithm conjecture. For every fixed $k\geq3$, it is polynomial-time decidable whether a tournament $T$ satisfies

$$\vec{\chi}_{\rm a}(T)\leq k.$$

The case $k=2$ is proved polynomial-time solvable in the paper, whereas the corresponding statement for every fixed $k\geq3$ is posed as an open complexity problem.