Bounded equitable vertex arborable threshold for planar graphs

Let $G$ be a simple finite planar graph, and let $va^{\equiv}(G)$ denote its equitable vertex arborable threshold: the minimum integer $k$ such that $G$ admits an equitable tree-$k'$-coloring for every integer $k'\geq k$. A tree-$k$-coloring is a coloring of $V(G)$ with colors $1,2,\dots,k$ such that each color class induces a forest, and the coloring is equitable when any two color-class sizes differ by at most one. Bounded planar equitable vertex arborable threshold conjecture. There is a constant $C$ such that

$$va^{\equiv}(G)\leq C$$

for every planar graph $G$. This asks whether the equitable vertex arborable threshold is bounded independently of the size and maximum degree of planar graphs; the source provides no evidence of resolution.