Equitable Vertex Arboricity Conjecture for bounded-degree graphs

Let $G$ be a graph with maximum degree at most $\Delta$. An equitable tree-$k$-coloring of $G$ assigns each vertex a color in $\{1,2,\ldots,k\}$ so that every color class induces a forest and the sizes of any two color classes differ by at most $1$. The equitable vertex arborable threshold $va_{eq}^*(G)$ is the least integer $k$ such that $G$ has an equitable tree-$k'$-coloring for every integer $k'\geq k$.

Equitable Vertex Arboricity Conjecture. Every graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$; equivalently,

$$va_{eq}^*(G)\leq \lceil(\Delta+1)/2\rceil.$$

Wu, Zhang and Li proposed this conjecture. It is known for all 5-degenerate graphs, and hence for graphs with maximum degree at most $5$, but remains open in general.