Erdős's bounded monochromatic reachability conjecture

Let $k$ be a positive integer. A $k$-edge-coloured finite tournament is a finite tournament whose edges are assigned one of $k$ colours, and a set $S\subseteq V(T)$ is reachable from a vertex by a monochromatic path if some monochromatic directed path from that vertex ends in a vertex of $S$. Erdős's conjecture. For every positive integer $k$ there is a least positive integer $f(k)$ such that every $k$-edge-coloured finite tournament admits a subset $S\subseteq V(T)$ of size at most $f(k)$ such that $S$ is reachable from every vertex by a monochromatic path. It is known that $f(1)=f(2)=1$ and that $f(3)\geq 3$, but no constant upper bound for $f(3)$ is known; Erdős conjectured that $f(3)=3$.