The Two-Colour-Conjecture for oriented planar digraphs

An oriented digraph is a digraph whose underlying undirected graph is simple, and a digraph is $k$-colourable if its vertices admit a colouring with $k$ colours such that every colour class is acyclic.

Two-Colour-Conjecture. Every oriented planar digraph $D$ is $2$-colourable.

This is the directed analogue of the Four-Colour Theorem. The conjecture was posed by Erdős and Neumann-Lara, and independently by Skrekovski, and remains open.