Minimal area of $\{|f|<1\}$ over polynomials rooted in $F$: zero when capacity $\ge 1$? (Erdős #1040)

Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of the area (two-dimensional Lebesgue measure) $$\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,$$ as $f$ ranges over all polynomials of the shape $\prod (z-z_i)$ with $z_i\in F$ (roots need not be distinct). Erdős, Herzog, and Piranian asked: is $\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\mu(F)=0$ whenever the transfinite diameter of $F$ is $\geq 1$? Here the transfinite diameter (equivalently, logarithmic capacity) of $F$ is $$\rho(F)=\lim_{n\to \infty}\sup_{z_1,\ldots,z_n\in F}\left(\prod_{i<j}\lvert z_i-z_j\rvert\right)^{1/\binom{n}{2}}.$$ The general question has recently been answered in the negative, and the case of transfinite diameter strictly greater than $1$ has since been resolved affirmatively (see background); what remains open is the boundary case of transfinite diameter exactly $1$: prove or disprove that $\mu(F)=0$ for every closed infinite $F$ with $\rho(F)=1$.

Acceptance. FULLY RESOLVES: a complete proof that $\mu(F)=0$ for every closed infinite $F\subseteq\mathbb{C}$ with transfinite diameter exactly $1$; OR a counterexample — a specific closed infinite set $F$ together with a proof that $\rho(F)=1$ and a proof that $\mu(F)>0$, i.e. a uniform positive lower bound on the area of $\{\lvert f\rvert<1\}$ over all polynomials with all roots in $F$. Both directions are proof-shaped (the quantities are infima over infinite families): a machine-checkable proof (Lean/Coq) is preferred, else a complete written proof with all steps. ADVANCES: a proof of $\mu(F)=0$ for structural classes of transfinite-diameter-exactly-$1$ sets strictly beyond the bounded $C^2$ smooth domains settled in [GhRa26] (e.g. all compact connected sets of capacity $1$, or all finite unions of smooth arcs of capacity $1$); quantitative bounds relating $\rho(F)$ to $\mu(F)$ near capacity $1$ that strictly improve those stated in the background; or a characterization, within a natural class of capacity-$1$ sets, of exactly which $F$ have $\mu(F)=0$. Deliver the proof file (Lean or full manuscript), and for any counterexample the capacity computation and the area lower-bound argument.