Bounded chromatic number for locally bounded tournaments

Let $t\geq 1$. A t-local tournament is a tournament $T$ such that, for every vertex $v$, the subtournament induced by the out-neighbors of $v$ has chromatic number at most $t$. The chromatic number $\vec \chi(T)$ is the minimum number of colors in a coloring of $T$ in which every color class is transitive.

Local-to-global tournament coloring conjecture. There is a function $f$ such that every $t$-local tournament $T$ satisfies

$$\vec \chi(T)\leq f(t).$$

The conjecture was raised in the cited work and was settled for $t=2$; the supplied text also states a theorem proving the same assertion, so its resolution should be checked against the paper's final version.