Let $F\subset\mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in any closed disc of radius $1$.

If $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ with all $z_i\in F$ then can

$$\{ z: \lvert f(z)\rvert < 1\}$$

have $n$ connected components?

If the transfinite diameter of $F$ is $<1$ then must this set only have at most $(1-c)n$ connected components, where $c>0$ depends only on $F$ (or just the transfinite diameter of $F$)?