Maximum-degree bound for equitable vertex arboricity

Let $G$ be a graph, let $\Delta(G)$ be its maximum degree, and let $va^{\equiv}_{\infty,\infty}(G)$ denote the strong equitable $(\infty,\infty)$-vertex-arboricity. Maximum-degree equitable vertex arboricity conjecture. For every graph $G$,

$$va^{\equiv}_{\infty,\infty}(G)\leq \left\lceil\frac{\Delta(G)+1}{2}\right\rceil.$$

The paper notes that the planar conjecture does not extend to general graphs, since some complete bipartite graphs have strong equitable vertex arboricity of order $\Omega(|G|^{1/2})$. This degree bound is proposed as a possible general-graph replacement and remains open in the supplied source.