Chromatic-polynomial separation conjecture for signed Petersen graphs

Let $P$ be the Petersen graph, let $\sigma$ range over its sign functions, and write $\chi_{(P,\sigma)}(k)$ and $\chi^*_{(P,\sigma)}(k)$ for the numbers of proper signed $k$-colorations and zero-free proper signed $k$-colorations, respectively. Switching isomorphism identifies the six switching-isomorphism classes of sign functions.

Chromatic separation conjecture. (a) Two signed Petersen graphs that are not switching isomorphic have different zero-free chromatic polynomials; in particular, they have different numbers $\chi_{(P,\sigma)}^*(4)$ of zero-free proper $2$-colorations. (b) For any $\mu\geq 2$, the six values $\chi_{(P,\sigma)}(2\mu+1)$ are different for the six switching-isomorphism classes of sign functions, and so are the six values $\chi^*_{(P,\sigma)}(2\mu)$.

The paper proves the analogous separation for ordinary chromatic polynomials and proper $1$-colorations, while the zero-free and higher-color claims are proposed as distinguishing properties and remain unproved in the supplied text.