The local-to-global conjecture for acyclic dichromatic number

For a tournament $T$, let $N^+(v)$ be the out-neighborhood of $v$, and let $\vec{\chi}_{\rm a}(T)$ denote its acyclic dichromatic number.

Local-to-global conjecture. There exists a function $f\colon \mathbb{N}\to\mathbb{N}$ such that every tournament $T$ satisfies

$$\vec{\chi}_{\rm a}(T)\leq\max_{v\in V(T)}f\bigl(\vec{\chi}_{\rm a}(T[N^+(v)])\bigr).$$

The analogous statement for the ordinary dichromatic number is known. The acyclic version is proposed as a stronger local-to-global property and remains open.