Traub's basin conjecture for polynomial root-finding maps

Let $p$ be a polynomial of degree $d\geq 2$. Assume that $p(\alpha)=0$, and let $T_{p,1}$ be the corresponding Traub's map. Denote by $\mathcal A^{\star}(\alpha)$ the basin of attraction of the root $\alpha$ with the root itself removed. Traub's basin conjecture. The set $\mathcal A^{\star}(\alpha)$ is simply connected and unbounded. The conjecture would extend the known basin results for Newton's method to Traub's method and would help construct a set of good initial conditions for finding all roots simultaneously. Its resolution is not established in the supplied source.