The bounded-chamber conjecture for signed reduced A-discriminants of bivariate 5-nomials

Let $(\mathcal{A},\varepsilon)$ be the signed support of a bivariate $5$-nomial and let

$$\bar{\xi}_{B,\varepsilon}\colon \{ \mu \in \mathbb{R} \mid \operatorname{sign}(B \begin{bmatrix}\mu \\ 1 \end{bmatrix}) = \varepsilon \} \to \mathbb{R}^2$$

be the parametrization map of $\Gamma_\varepsilon(A,B)$ as defined in the paper. If $\bar{\xi}_{B,\varepsilon}$ has two critical points, then the complement of $\Gamma_\varepsilon(A,B)$ has a bounded connected component.

Bounded-chamber conjecture. Whenever the parametrization map has two critical points, the complement of the signed reduced $A$-discriminant contains a bounded connected component.

The conjecture records the pattern observed in experiments for bivariate $5$-nomials. The paper states that no proof or counterexample is known.