Neumann–Lara's 2-colourability conjecture for planar digraphs

Let $D$ be an oriented planar graph, meaning a planar digraph without directed cycles of length at most $2$. A $2$-colouring of $D$ is a function $f:V(D)\to\{1,2\}$ such that the subdigraph induced by the vertices of each colour is acyclic. Neumann–Lara's conjecture. Every oriented planar graph is $2$-colourable. The conjecture was proposed by Neumann–Lara in 1985 and independently by Škrekovski. The paper proves the relaxed version for planar digraphs of digirth at least four, while the case of digirth at least three remains open.