The chromatic pairs polynomial refines the chromatic polynomial

Let $G_1$ and $G_2$ be graphs, and let $\pi_{G_i}^{(P_2)}(k)$ denote their chromatic pairs polynomials and $\pi_{G_i}(k)$ their chromatic polynomials. Chromatic-pairs refinement conjecture. If two graphs have the same chromatic pairs polynomial, then they have the same chromatic polynomial.

This conjecture proposes that the chromatic pairs polynomial is a graph invariant at least as discriminating as the chromatic polynomial. It is proved for trees in the paper, while the general case remains open.