Henrot's reverse Faber–Krahn conjecture for the Steklov-type eigenvalue

Let $\Omega\subset\mathbb R^n$ be an open bounded Lipschitz set, and let $\Omega^\sharp$ denote a ball with the same measure as $\Omega$. For such a domain, define

$$\lambda(\Omega)=\min_{\substack{w\in H^1(\Omega)\w\ne0}}\frac{\displaystyle\int_\Omega |Dw|^2\,dx+\int_\Omega w^2\,dx}{\displaystyle\int_{\partial\Omega}w^2\,d\mathcal H^{n-1}}.$$

Henrot's conjecture. One has

$$\lambda(\Omega)\le\lambda(\Omega^\sharp),$$

and therefore, among open bounded Lipschitz sets of given measure, the ball achieves the worst (least) embedding constant in the Sobolev–Poincaré trace inequality $\|w\|_{L^2(\partial\Omega)}\le C\|w\|_{H^1(\Omega)}$. This is a reverse Faber–Krahn-type assertion for the Steklov-type eigenvalue and was proposed in analogy with the Brock–Weinstock inequality; the source gives no resolution evidence for this formulation.