The universal quadratic refinement of Sendov's conjecture

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$, and let $d(P,\beta)$ be the distance from $\beta$ to the closest root of $P'$. The universal quadratic refinement conjecture. There is a constant $c\approx0.233$ such that, for every $P\in S(\beta)$,

$$d(P,\beta)\le1-c\beta(1-\beta).$$

This is the paper's experimentally motivated conjecture for arbitrary degree and arbitrary root configurations, and would provide a uniform strengthening of Sendov's conjecture. It remains open.