0-coloring for planar graphs of girth at least four

Prove that every planar graph of girth at least 4 admits a \((0)\)-coloring, that is, a vertex partition into an independent set and a forest.

Background

The paper records a conjecture of Dross, Montassier, and Pinlou concerning a strengthening of the known result that every planar graph of girth at least 5 admits a $(0)$-coloring.

The conjectured extension lowers the girth requirement from 5 to 4. The paper later discusses stronger positive results for planar graphs of girth at least 4 with additional restrictions, but does not establish the stated unrestricted conjecture.