Planar equitable vertex arboricity conjecture

Let $G$ be a planar graph, and let $va_{eq}^*(G)$ denote the least integer $k$ such that $G$ has an equitable tree-$k'$-coloring for every integer $k'\geq k$, where an equitable tree-$k'$-coloring has forest color classes whose sizes differ pairwise by at most $1$.

Planar equitable vertex arboricity conjecture. There exists a constant $k$ such that every planar graph is equitably tree-$k'$-colorable for every integer $k'\geq k$; equivalently,

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

This conjecture was proposed by Wu, Zhang and Li and was later confirmed by Esperet, Lemoine and Maffray, who proved the stronger explicit bound $va_{eq}^*(G)\leq 4$ for every planar graph. Thus the conjecture is solved.