The Erdős–Sands–Sauer–Woodrow path-domination conjecture

Let a tournament have its edges colored with $k$ colors. A set $S$ of vertices has the path-domination property if every vertex outside $S$ can be reached from some vertex of $S$ by a monochromatic directed path.

Erdős–Sands–Sauer–Woodrow conjecture. For each positive integer $k$ there is a least integer $f(k)$ such that every $k$-colored tournament contains a set $S$ of $f(k)$ vertices with the path-domination property. In particular, does $f(3)$ exist, and is $f(3)=3$?

The paper identifies this as a well-known conjecture and cites subsequent work for further developments. The provided text does not state that it has been resolved.