Attainment conjecture for the conformal Steklov invariant

Let $\Sigma$ be a compact surface, and let $I^\sigma_k(\Sigma)$ denote the infimum of the normalized $k$-th Steklov eigenvalue over conformal classes on $\Sigma$. Attainment conjecture. The infimum $I^\sigma_k(\Sigma)$ is attained if and only if $\Sigma$ is diffeomorphic to the disc $\mathbb D^2$. This conjecture concerns when the conformal Steklov infimum is realized; the source presents it as open and plans to address it in subsequent work.