Extremal Steklov domain symmetry and uniqueness conjecture

Let $\Omega\subset\mathbb R^2$ be a domain, let $|\Omega|$ denote its area, and let $\lambda_p(\Omega)$ be its $p$-th Steklov eigenvalue. Define the scale-invariant quantity

$$\Lambda_p(\Omega)=\lambda_p(\Omega)\sqrt{|\Omega|},\qquad \Lambda^{p\star}=\max_{\Omega\subset\mathbb R^2}\Lambda_p(\Omega).$$

Extremal Steklov domain conjecture. The maximizer $\Omega^{p\star}$ of $\Lambda_p(\Omega)$ is unique up to dilations and rigid transformations, has $p$-fold symmetry and an axis of symmetry, and the $p$-th Steklov eigenvalue has multiplicity $2$ if $p$ is even and multiplicity $3$ if $p\geq 3$ is odd. This conjecture is based on computational experiments for $p$ between $1$ and $101$; the stated uniqueness, symmetry, and multiplicity properties remain conjectural in the source.