Erdős–Neumann-Lara conjecture on chromatic and dichromatic numbers

Let $G$ be a graph. An orientation of $G$ is a digraph with underlying graph $G$ and at most one arc between each pair of vertices. For a digraph $D$, its dichromatic number $vec\chi(D)$ is the minimum number of colours in a vertex colouring whose colour classes induce acyclic subdigraphs; define

$$\vec\chi(G)=\max_D\vec\chi(D),$$

where the maximum is over all orientations $D$ of $G$. Let $\chi(G)$ denote the chromatic number of $G$.

Erdős–Neumann-Lara conjecture. For every integer $k$ there is an integer $f(k)$ such that, for every graph $G$, $\chi(G)\geq f(k)$ implies $\vec\chi(G)\geq k$.

This asks whether arbitrarily large chromatic number forces an orientation with arbitrarily large dichromatic number. The problem is open: $f(1)=1$ and $f(2)=3$ are possible, but it is unclear whether $f(3)$ exists.