The sharp first Steklov eigenvalue conjecture for planar domains

Let $\widehat \sigma_1^*$ denote the supremum of the first nonzero Steklov eigenvalue multiplied by the boundary length, over planar domains of area $\pi$, and let $\mu_1(\mathbb{D})$ be the first nonzero Neumann eigenvalue of the unit disk $\mathbb{D}$. Sharp first Steklov eigenvalue conjecture.

$$\widehat \sigma_1^* = \mu_1(\mathbb{D})\times\pi\approx 3.39\pi.$$

The preceding homogenisation construction shows that the value on the right can be approached arbitrarily closely by pierced unit disks with sufficiently large parameter $\beta$. Thus the conjecture asserts that this lower bound is sharp among planar Euclidean domains; the supplied context does not indicate whether it has been resolved.