Connectedness of extremal Green level sets

Definitions and mathematical statement

For a compact set $E\subset\mathbb C$ of positive logarithmic capacity, let $G_E$ be the Green function with pole at infinity of the unbounded complementary component, extended by zero elsewhere. Assume regularity, meaning that this extension is continuous. Normalize $G_E(z)-\log|z|\to-\log\operatorname{cap}(E)$ as $z\to\infty$. Define \[ \beta_E=\limsup_{\varepsilon\downarrow0} \frac{\log\mathcal H^1\{z:G_E(z)=\varepsilon\}} {-\log\varepsilon}, \] where $\mathcal H^1$ measures the total length of all level-curve components.

Compare $\beta_{\mathrm{all}}=\sup_E\beta_E$, over all such regular compact sets, with $\beta_{\mathrm{conn}}=\sup_{E\ \mathrm{connected}}\beta_E$. Is the unrestricted supremum attained on connected sets? In particular, determine whether $\beta_{\mathrm{all}}=\beta_{\mathrm{conn}}$, so that allowing disconnected compact sets does not increase the optimal exponent.

Distinguish this equality of suprema from the stronger attainment question: does there exist one regular connected compact set $E_*$ with $\beta_{E_*}=\beta_{\mathrm{all}}$? The source's wording invites investigation of both. A comparison of suprema may be established by connected examples approaching every unrestricted exponent without producing a single maximizer. Disconnected sets may have arbitrarily many components. No restriction on component size, separation, boundary regularity beyond the Green-function condition, or finite boundary length is imposed.

Short English statement

Does permitting disconnected compact sets increase the largest Green-level length exponent, and can a connected set attain the optimum?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700034 (AMR-036-0034), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Littlewood constants, Section 2 second conjecture. Original source indexed by AMR-036-0034. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Some constants coming from the work of Littlewood, Section 2, connectedness question. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/lit.pdf.