The site labels the problem PROVED (LEAN); the Lean suffix refers to a formal proof by others, recorded on the Pommerenke claim page, that this corpus has not built or audited. The site credits the answer yes to Pommerenke [Po61], who proves that the set contains a disk of radius $(2e)^{-1}n^{-2}$, so its area is at least a constant times $n^{-4}$, in a paper refereed in the Michigan Mathematical Journal; the claim page is Pommerenke 1961. The parenthetical stronger bound, an area of at least $(\log n)^{-O(1)}$, is proved with exponent $1$ by Krishnapur, Lundberg and Ramachandran [KLR25] in a preprint the site credits; the claim page is Krishnapur, Lundberg and Ramachandran 2025. See Current assessment for the evidence.