Linear starting-point conjecture for boundary roots of Newton's method

Let $p$ be a polynomial of degree $d$, rescaled so that all its roots lie in the unit disk. For $r>1$, let $\partial D_r(0)$ be the circle of radius $r$ centered at the origin, and let equidistributed points be equally spaced on this circle. Starting-point conjecture for boundary roots. For every radius $r>1$, there is a universal constant $c>1$ such that $\lceil cd\rceil$ equidistributed points on $\partial D_r(0)$ have the property that, for each root on the boundary of the convex hull of all roots, at least one of the points converges to that root under iteration of Newton's method. The conjecture predicts a universal set of $O(d)$ starting points for finding all roots lying on the boundary of the convex hull; the authors believe that $c\approx 2.6$ for large $r$.