Many derivative-root circles conjecture for Newton basins

Let $p$ be a degree-$n$ univariate complex polynomial. Let its DR-circles be the circles centered at the roots of $p'$ with radii $\rho_j$, where

$$\rho_j=\min_{k=2,\ldots,n}\left|\frac{p(\zeta_j)k!}{p^{(k)}(\zeta_j)}\right|^{1/k}.$$

For each root of $p$, consider its basin of fast convergence under Newton's method. Many derivative-root circles conjecture. There exist universal constants $\eta_2>0$ and $\eta_3>0$ such that at least $\eta_2n$ DR-circles contain segments of length $\eta_3$ radians lying in the union of the basins of fast convergence of the roots of $p$. This would strengthen the algorithmic usefulness of the DR-circles by asserting that a linear number of them contain constant-sized useful segments. The supplied material gives no resolution status.