Grünbaum's equitable coloring conjecture

Let $G$ be a graph with maximum degree $\Delta(G)$, and let a $k$-equitable coloring be a proper coloring with $k$ color classes whose sizes differ by at most one. Grünbaum's conjecture. Every graph $G$ with

$$\Delta(G)<k$$

has a $k$-equitable coloring. The conjecture was proved by Hajnal and Szemerédi in 1970.