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Claim. For every $c\in(0,4)$ and every $N$ there is a monic polynomial $p$ of some degree $n$ such that $\{z:\lvert p(z)\rvert\le1\}$ has at least $N$ connected components each of diameter at least $c$. This is Theorem 1.1 of L. Huang, Many lemniscates with large diameter, arXiv:2509.11597, first posted 15 September 2025, with a source card in the library. The method (Sections 2.1--2.3) builds one explicit domain $\Omega$, bounded by the image of the circle $\{\lvert z\rvert=4/c\}$ under a shifted Joukowski map, which contains the segment $[0,c]$ and has logarithmic capacity exactly $1$; places inside $\Omega$ a set $\Omega_N$ of $N$ pairwise disjoint Jordan domains, each of diameter greater than $c$, separated from one another and from the boundary of $\Omega$; and applies the Hilbert lemniscate theorem to $\Omega_N$ with an $\varepsilon$-neighborhood that has $N$ separated components and lies inside $\Omega$, obtaining a polynomial $q$ with $\Omega_N\subset\{\lvert q\rvert\le1\}\subset\Omega$ after normalization. Since $\operatorname{Cap}(q^{-1}(\overline{\mathbb D}))\le\operatorname{Cap}(\Omega)=1$ and the capacity of $r^{-1}(\mathbb D)$ is $\lvert a_d\rvert^{-1/d}$ for a polynomial $r$ of degree $d$ with leading coefficient $a_d$, the leading coefficient of $q$ has modulus at least $1$, and $p(z)=q(z/w)$ with $w^d=a_d$ is monic with $p^{-1}(\overline{\mathbb D})=w\cdot q^{-1}(\overline{\mathbb D})$, a rotation and dilation by $\lvert w\rvert\ge1$ that can only enlarge diameters. The author notes that the restriction $c<4$ is optimal by Pólya's projection bound. The theorem answers the question of Problem 511 in the negative, and the author adds that the problem had already been solved by Pommerenke, whose claim page records the 1961 theorem; the two constructions are independent. The paper states its theorem for the closed sublevel set, while the problem is posed for the open set $\{\lvert p\rvert<1\}$.

Acceptance. The site's curator, Thomas Bloom, labels the problem disproved and credits this paper as an independent proof of the negative answer, the reviewed evidence. The paper is an arXiv preprint and no journal publication of it has been identified, so no refereed evidence is listed.

Depends on. Nothing beyond the cited paper.