Erdős problem 1044

Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of $$\{ z: \lvert f(z)\rvert<1\}$$ then determine the infimum of $\Lambda(f)$.

A problem of Erdős, Herzog, and Piranian [EHP58].

This has been resolved by Tang, who proved that the infimum of $\Lambda(f)$ over all such $f$ is $2$.