According to this link (see point $15$), Matt DeVos and David Wood can prove that every digraph in which each vertex has an out-neighbour has a good $4$-colouring (I am using their terminology).
Update. The paper Majority Colourings of Digraphs by Kreutzer, Oum, Seymour, van der Zypen (the OP), and Wood is now available on the arXiv. They give a short proof for $4$ colours and provide evidence for the conjecture with $3$ colours.