Three-region zero-separation conjecture for the electrostatic polynomial sequence

Let $G(d;z)$ be the polynomial considered in the paper, let $A_1,A_2,A_3$ be the three sets used to organize its zeros, and let

$$\mathcal C_0=\{z:|z|=1\}$$

be the unit circle. For $6\mid d$, also consider the two exceptional zeros $(1\pm\mathrm{i}\sqrt{3})/2$.

Zero-separation conjecture. For every positive integer $d$, the zeros of $G(d;z)$ form three groups separated by the sets $A_1$, $A_2$, and $A_3$, except that when $6$ divides $d$ one also has the zeros $(1\pm\mathrm{i}\sqrt{3})/2$.

This conjecture concerns the global organization of the zeros and complements the preceding asymptotic description of their attracting set. The paper presents it among challenging conjectures, with no proof or resolution supplied.