The signed-threshold conjecture for bivariate chromatic polynomials

For each integer $n\geqslant 0$ and each vector $\bm{a}\in\{-1,0,1\}^n$, let $\mathcal{T}_{\bm{a}}$ be the associated signed threshold graph, with bivariate chromatic polynomial $\mathscr B(\mathcal{T}_{\bm{a}},x,y)$ and chromatic polynomial $\mathsf E(\mathcal{T}_{\bm{a}},x,y)$. Signed-threshold conjecture. Let $n \geqslant 0$ be an integer and let $\bm{a}, \bm{b} \in \{-1,0,1\}^n$. Then

$$\bm{a} = \bm{b} \iff \mathcal{T}_{\bm{a}} = \mathcal{T}_{\bm{b}} \iff \mathscr B(\mathcal{T}_{\bm{a}},x,y) = \mathscr B(\mathcal{T}_{\bm{b}},x,y) \iff \mathsf E(\mathcal{T}_{\bm{a}},x,y) = \mathsf E(\mathcal{T}_{\bm{b}},x,y).$$

The conjecture extends an established result for the parameter sets $\{0,1\}^n\cup\{\pm1\}^n$ to $\{-1,0,1\}^n$. The source reports computational verification through $n=12$, but the general assertion remains open.