Strict conformal Steklov rigidity conjecture for non-discs

Let $\Sigma$ be a compact surface, let $c$ be a conformal class on $\Sigma$, and let $\sigma^*_1(\Sigma,c)$ denote the normalized first Steklov eigenvalue in that conformal class. Strict conformal Steklov rigidity conjecture. If $\Sigma$ is not diffeomorphic to the disc, then for every conformal class $c$ on $\Sigma$ one has

$$\sigma^*_1(\Sigma,c)>\sigma^*_1(\mathbb D^2)=2\pi.$$

This is presented as an analogue of the Petrides rigidity theorem. It is known for the cylinder and the Möbius band, but remains open in general.