Montgomery's dynamic chromatic number conjecture for regular graphs

Let $G$ be a regular graph. Its chromatic number is denoted by $\chi(G)$, and its dynamic chromatic number, the smallest number of colors in a dynamic proper vertex coloring, is denoted by $\chi_2(G)$. Montgomery's conjecture.

$$\chi_2(G)-\chi(G)\leq 2.$$

The difference between dynamic chromatic number and chromatic number can be arbitrarily large for general graphs, so the regularity hypothesis is essential. The conjecture is resolved by the result cited in the source.