Erdős problem 1039

For $f(z) = \prod_{i=1}^n (z - z_i)$ with all $|z_i| \le 1$, let $\rho(f)$ be the radius of the largest disc contained in $\{z : |f(z)| < 1\}$. Is $\rho(f) \gg 1/n$? The worst case is now known to be $\Theta(1/n)$, with the explicit bound $\rho(f) \ge (\log 2)/n$.