The low-degree extremal values conjecture for Sendov's refinement

For $n\ge2$, let $r_n(\beta)$ be the maximum of $d(P,\beta)$ over $P\in S(\beta)$ of degree $n$, and let $c_n$ denote the corresponding sharp quadratic-refinement coefficient in the bound $d(P,\beta)\le1-c_n\beta(1-\beta)$. The experimentally conjectured values are listed in Table 2: $c_4=1/3$, $c_5=3/10$, $c_6=0.365121611819106$, $c_7=0.335088765359222$, and $c_8=(8-4\sqrt2)/7\approx0.334735107215374$. The low-degree extremal values conjecture. The conjectured values of $c_n$ listed in Table 2 are the correct values. These values would substantially improve the bounds supplied by Sendov's conjecture for the indicated degrees, but the assertion is based on experimental data and remains open.