Xu–Liu conjecture on degree sequences of chromatically equivalent complements

Let $G$ be a simple graph, let $p_G(k)$ denote its chromatic polynomial, and let $\overline{G}$ denote its complement. Suppose that

$$p_G(k)=p_{\overline{G}}(k).$$

Xu–Liu conjecture. Then $G$ and $\overline{G}$ have the same degree sequence. This conjecture concerns graphs whose chromatic polynomial agrees with that of their complement. The paper disproves it by constructing infinitely many graphs with the stated chromatic-polynomial property but different degree sequences from their complements.