Miller's quadratic refinement conjecture for Sendov's problem

Let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the unit disk and at least one root at $\beta$. For $P\in S(\beta)$, let $d(P,\beta)$ be the distance between $\beta$ and the closest root of $P'$, and define

$$r(\beta)=\sup\{d(P,\beta):P\in S(\beta)\}.$$

By rotation, assume $0\leq\beta\leq 1$. Miller's quadratic refinement conjecture. For every $\beta\in[0,1]$,

$$r(\beta)\leq 1-\frac{3}{10}\beta(1-\beta).$$

This strengthens Sendov's conjectured bound while preserving equality at the endpoints $\beta=0$ and $\beta=1$. The paper proves it for polynomials of degree $2$ or $3$, for real polynomials of degree $4$, and in several further special cases, but leaves the general assertion open.