Many lemniscates with large diameter

We prove that for every $0<c<4$ and every $N\in \mathbb{N}$ there exists a monic polynomial $p(z)=z^n+a_{n-1}{z}^{n-1}+\cdots +a_0$ such that the set $\{z\in \mathbb{C}:\mathrm{\mid }p(z)\mathrm{\mid }\le 1\}$ has at least $N$ connected components with diameter at least $c$. This answers a question of Erdős.