The conjecture on separated boundary components for Steklov maximizers

Let $\Sigma$ be a surface with disconnected boundary $\partial\Sigma$, and let $g$ be a maximizer of the normalized first Steklov eigenvalue $\bar{\sigma}_1(\Sigma,\cdot)$. Define $\Delta(\partial\Sigma)$ to be the set of pairs $(p,q)$ such that $p$ and $q$ belong to disjoint connected components of $\partial\Sigma$. Let $\mathcal{C}(\Sigma,g)$ be the set of maps

$$\Phi:(\Sigma,g)\to(\mathbb{B}^n,\mathbb{S}^{n-1})$$

satisfying

$$\Delta_g\Phi=0\quad\text{in }\Sigma,$$ $$\partial_\nu\Phi=\sigma_1(\Sigma,g)\cdot\Phi\quad\text{on }\partial\Sigma,$$

and which are possibly branched conformal maps. Separated-boundary Steklov conjecture. There exists $(p,q)\in\Delta(\partial\Sigma)$ such that, for every $\Phi\in\mathcal{C}(\Sigma,g)$, one has $\Phi(p)\neq\Phi(q)$. This conjecture is intended to provide the Steklov analogue of the corresponding existence and monotonicity results for closed surfaces. It is stated as true when $\Sigma$ has genus $0$, while the general case for surfaces with disconnected boundary is left as an open question.