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Manjunath Krishnapur, Erik Lundberg and Koushik Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270 (24 March 2025), prove a regular case of the second question. For a compact $K$ let $\kappa_n(K,1)$ be the minimal area of $\{z:|p(z)|<1\}$ over monic $p$ of degree $n$ with all zeros in $K$. Theorem 6 states that if $K$ is the closure of a bounded open set with $C^2$-smooth boundary and has capacity $1$, then $\inf_n\kappa_n(K,1)=0$, which is $\mu(K)=0$ in the problem's notation; Theorem 7 adds that for any compact $K$ of capacity $1$, zeros of polynomials whose sublevel areas in $K$ tend to zero equidistribute to the equilibrium measure of $K$. For the closed unit disc, the case Erdős, Herzog and Piranian had settled qualitatively, the paper proves $c/\log n\le\kappa_n(\overline{\mathbb D},1)\le C/\log\log n$, improving Pommerenke's lower bound and Wagner's upper bound. Its introduction records that the case of capacity above $1$ had been treated under a regularity condition in the authors' earlier paper, Inradius of random lemniscates, J. Approx. Theory 299 (2024), Paper No. 106018, whose Corollary 1.6 gives $\mu(K)=0$ for compact $K$ of capacity above $1$ under a Frostman-type bound on the equilibrium measure (recorded on its page), and that the general unit-capacity case loses quantitative control. The paper's library card records it; the statements above are those of the arXiv version, and the proofs are not checked here.

Covers. The second question for one class of sets: a compact $K$ of capacity $1$ that is the closure of a bounded open set with $C^2$ boundary has $\mu(K)=0$. The claim says nothing about other compact sets of capacity $1$, about unbounded closed sets, or about the first question; the case of capacity above $1$ is recorded on [[problems/analysis/E1040/claims/2026_04_03_ghosh_ramachandran|the page of Ghosh and Ramachandran]], and the general case on the four later pages of the folder.

Standing. The arXiv record lists no journal reference (checked 2026-10-07), the paper is not named on the site's page or thread, and no reviewer is named, so the claim stays claimed.