In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for every $Δ\in \mathbb{Z}_{\geq 3}$ there exists a complex neighborhood of the interval $\left[0, \frac{\left(Δ- 1\right)^{Δ- 1}}{\left(Δ-2\right)^Δ}\right)$ on which the independence polynomial is nonzero for all graphs of maximum degree $Δ$. Furthermore, they gave an explicit neighborhood $U_Δ$ containing this interval on which the independence polynomial is nonzero for all finite rooted Cayley trees with branching number $Δ$. The question remained whether $U_Δ$ would be zero-free for the independence polynomial of all graphs of maximum degree $Δ$. In this paper it is shown that this is not the case. [1] Han Peters and Guus Regts, On a conjecture of sokal concerning roots of the independence polynomial, Michigan Math. J. (2019), Advance publication.