Connectedness conjecture for the complement of the zero-locus

For each integer $\Delta\geq 3$, let $\mathcal{G}_\Delta$ be the class of graphs of maximum degree at most $\Delta$, and let

$$\mathcal{Z}_\Delta=\{\lambda\in\mathbb{C}:Z_G(\lambda)=0\text{ for some }G\in\mathcal{G}_\Delta\},$$

where $Z_G(\lambda)$ is the independence polynomial of $G$ evaluated at $\lambda$. The connectedness conjecture. For each integer $\Delta\geq 3$, the set

$$\mathbb{C}\setminus\overline{\mathcal{Z}_\Delta}$$

is connected. The topology of the complement of the zero-locus is not understood; resolving this conjecture would clarify the global structure of the zero-free region relevant to approximating the independence polynomial.