The two-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 two-critical-point refinement conjecture. If $P\in S(\beta)$ has at most $2$ distinct critical points, then

$$d(P,\beta)\le1-\frac{3}{10}\beta(1-\beta).$$

The coefficient is motivated by numerical experiments and the paper's results for the cases where the degree is or is not divisible by $3$. The general assertion for this class remains open.