Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1$, the set$$\{z\in \mathbb{C}:|f(z)|<1\}$$has at most $O_c(1)$ many connected components of diameter $>c$ (where the implied constant is in particular independent of $n$)?