A quadratic refinement of Sendov's conjecture

Let $\beta$ be a real number in $[0,1]$. For a polynomial $P$ of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$, let $d(P,\beta)$ be the distance from $\beta$ to the closest root of $P'$, and define

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

where $S(\beta)$ is the set of such polynomials. The quadratic refinement of Sendov's conjecture. There is a constant $c>0$ such that

$$r(\beta)\le 1-c\beta(1-\beta)\qquad\text{for all }\beta\in[0,1].$$

This strengthens Sendov's conjectured bound $r(\beta)\le1$ by requiring a uniform positive improvement away from the endpoints. The paper verifies the bound for several special classes and low degrees, but the general assertion remains open.