The four-critical-point refinement conjecture for 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 four-critical-point refinement conjecture. There is a constant $c\approx0.24483$ such that, if $P\in S(\beta)$ has at most $4$ distinct critical points, then

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

The numerical value is inferred from experimental searches for high-degree extremal polynomials. Establishing a uniform coefficient of this size for the whole class remains open.