The strict Steklov inequality for non-disk surfaces

Let $(\Sigma,g)$ be a compact Riemannian surface with boundary that is not diffeomorphic to the disk. Let $\sigma_1(\Sigma,[g])$ denote its first normalized Steklov eigenvalue, and let $\sigma_1(\mathbb{D})$ denote the corresponding value for the disk. Strict Steklov inequality conjecture.

$$\sigma_1(\Sigma,[g])>2\pi=\sigma_1(\mathbb{D}).$$

This is described as the simplest strict inequality needed in the surrounding existence theory, but it remains a conjecture in the source.