Kreutzer et al.'s majority 3-colouring conjecture for digraphs

Let $D$ be a digraph. A majority colouring of $D$ is a vertex colouring $c$ such that at least half of the out-neighbours of every vertex $v$ have a colour different from $c(v)$. Kreutzer et al.'s conjecture. Every digraph has a majority $3$-colouring. Kreutzer et al. proved that every digraph has a majority $4$-colouring, and established the conjecture for digraphs satisfying certain sufficiently large out-degree or bounded in-degree conditions; the general case remains open.