The gap-cycle conjecture for 2-coloring simple planar digraphs

Let $D$ be a simple planar digraph. Say that $D$ has no cycles of lengths $4,\dots,k$ when it contains no directed cycles whose lengths are among those integers. Gap-cycle conjecture. There exists $k$ such that every simple planar digraph without cycles of length $4,\dots,k$ is 2-colorable. This is presented as a relaxation of the Neumann-Lara–Škrekovski conjecture; the source says that even the original conjecture remains out of reach, so this question is open.