Planar equitable tree-3-coloring conjecture

Let $G$ be a planar graph. An equitable tree-$3$-coloring is a coloring of $V(G)$ with three colors such that each color class induces a forest and any two color classes differ in size by at most $1$. Write $va_{eq}^*(G)$ for the least integer $k$ such that $G$ has an equitable tree-$k'$-coloring for every integer $k'\geq k$.

Planar equitable tree-3-coloring conjecture. Every planar graph is equitably tree-$3$-colorable; equivalently,

$$va_{eq}^*(G)\leq 3.$$

The preceding bound $va_{eq}^*(G)\leq 4$ for planar graphs leaves only this sharper case. The source presents it as a natural conjecture, and no resolution is supplied there.