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Tony Feng and twenty-three coauthors, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401 (v1 of 29 January 2026, v3 of 5 February 2026), Section 3.2 (Section 3.3 of v1), answers the first question negatively. With $\mu(F)$ the infimum of the areas of $\{z:|f(z)|<1\}$ over monic $f$ with all zeros in $F$, the paper takes $F_1=\{0\}\cup\{1/n:n\ge1\}$ and $F_2=\{0,R\}\cup\{1/n:n\ge1\}\cup\{R+1/n:n\ge1\}$ with $R>4$. Both are countable compact sets, so both have transfinite diameter $0$; the paper shows $\mu(F_1)\ge\pi/4$, while $f(z)=z(z-R)$ gives $\mu(F_2)\le2\pi/(R^2-4)$, which is below $\pi/4$ once $R$ is large. Two closed infinite sets of equal transfinite diameter thus have different values of $\mu$. The paper reports the argument as found by its research agent Aletheia, built on Gemini, and classifies the problem as a partial AI solution: its Remark 3.2 (Remark 3.4 of v1) says that the second question was omitted because the model's reduction of it was incorrect, and a footnote by the authors notes that the model's output neither justified nor cited its general claim that every countable compact set has transfinite diameter (logarithmic capacity) zero, refers that claim to Ransford's Corollary 3.2.5, and adds that both constructed sets are easily checked directly. The paper's library card records the construction; the proof is not checked here.

Covers. The first question only: $\mu(F)$ is not determined by the transfinite diameter of $F$. The claim says nothing about the second question, whether $\mu(F)=0$ whenever the transfinite diameter of $F$ is at least $1$, which the paper explicitly leaves aside.

Standing. The site's commentary (last edited 1 February 2026) credits Aletheia [Fe26] with the negative answer to the first question and names both sets and both bounds, but the site labels the problem OPEN, so the commentary is not acceptance. The arXiv record lists three versions and no journal reference, and no outside reviewer is named, so the claim stays claimed. A second, elementary family of counterexamples at every capacity in $(0,1)$ is recorded on the page of Ghosh and Ramachandran.