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Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$ and \[A = \{ z\in \mathbb{C} : \lvert f(z)\rvert\leq 1\}.\] Is it true that, for every such $f$ and constant $c>0$, the set $A$ can have at most $O_c(1)$ many components of diameter $>1+c$ (where the implied constant is in particular independent of $n$)?