The chromatic-polynomial characterization of signed complete graphs

Let $\Sigma_1$ and $\Sigma_2$ be signed complete graphs. Write $\Sigma_1\cong\Sigma_2$ when they are isomorphic, and let $\mathscr B(\Sigma,x,y)$ and $\mathsf E(\Sigma,x,y)$ denote their bivariate and ordinary chromatic polynomials, respectively. Signed complete-graph characterization conjecture.

$$\Sigma_1 \cong \Sigma_2 \iff \mathscr B(\Sigma_1,x,y) = \mathscr B(\Sigma_2,x,y) \iff \mathsf E(\Sigma_1,x,y) = \mathsf E(\Sigma_2,x,y).$$

Thus, within signed complete graphs, both polynomials are conjectured to determine the graph up to isomorphism. The source gives this as a conjecture and does not provide a resolution.