NP-completeness of planar linear arboricity at maximum degree four

Let $G$ be a planar graph with maximum degree $4$. Its linear arboricity ${\rm la}(G)$ is the minimum number of linear forests whose union is the edge set of $G$.

Planar degree-four linear arboricity complexity conjecture. It is NP-complete to determine whether

$${\rm la}(G)=2.$$

The paper identifies this as the only remaining case in its discussion of planar graphs: the cases of odd maximum degree are immediate, maximum degree at least $9$ are settled by the main theorem, and the planar linear arboricity conjecture would settle maximum degrees $6$ and $8$.