Kostochka–Nakprasit equitable coloring conjecture for d-degenerate graphs

Let $\Delta$ be an integer, let $d\in\{2,\dotsc,\Delta\}$, and let $G$ be a $d$-degenerate graph with maximum degree at most $\Delta$. An equitable coloring is a proper vertex coloring in which the color classes have sizes differing by at most one. Kostochka–Nakprasit's conjecture. The graph $G$ admits an equitable $k$-coloring whenever

$$k\geqslant\frac{\Delta+d+1}{2}.$$

This generalizes the authors' earlier conjecture for $2$-degenerate graphs and would extend equitable-coloring results known for series-parallel and outerplanar graphs. The source presents it as an open problem; the stated theorem for $K_4$-minor-free graphs does not settle the conjecture for all $d$-degenerate graphs.