Pawlowski's asymptotic conjecture for the critical-point radius

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$ satisfying $|z_k|\leq 1$, and critical points $w_1,\dots,w_{n-1}$. Let $\gamma_n$ be the radius of the smallest disk centered at $\frac{1}{n}\sum_{k=1}^n z_k$ that contains at least one critical point of $F$. Pawlowski's conjecture. A sharp upper bound for $\gamma_n$ should be asymptotically of the form

$$\gamma_n\leq 1-c\frac{\log n}{n}$$

for some positive constant $c$. Pawlowski's existing upper bound has a weaker asymptotic correction of order $(\log n/n)^2$; the paper does not report a resolution of this sharper asymptotic claim.