The global annulus conjecture for the first normalized Steklov eigenvalue

Let $\Lambda_{\epsilon}$ be a bounded connected planar annulus with two boundary components, and let $\lambda_{1}(\Lambda_{\epsilon})$ be its first nontrivial Steklov eigenvalue, given by the Steklov eigenvalue formula for annuli. Global annulus conjecture. Among bounded connected planar domains with two boundary components, the best planar annulus should be the one that realizes the maximum of

$$\lambda_{1}(\Lambda_{\epsilon})|\partial\Lambda_{\epsilon}|.$$

The paper identifies the annulus with inner radius $\epsilon_{0}$ from its preceding critical-domain proposition as a candidate for a global maximizer up to scaling, based on numerical evidence; the global claim remains unproved in the source.