Equality characterization for the planar boundary-data eigenvalue inequality

Let $\Omega$ be a convex domain in $\mathbb{R}^2$ such that $\kappa_1=\mu_2$. Here $\kappa_1$ denotes the first eigenvalue of the variational problem defined in the paper, $\mu_2$ is the second Neumann Laplacian eigenvalue, $u_c$ is the solution associated with the boundary parameter $c$, $P(\Omega)$ is the perimeter, and $|\Omega|$ is the area. Planar equality conjecture.

$$\lim_{c\rightarrow \mu_2} c\int_{\partial \Omega}u_c\,d\sigma\geq \genfrac{}{}{}{}{1}{2}\genfrac{}{}{}{}{P^2(\Omega)}{|\Omega|},$$

with equality if and only if $\Omega$ is a disk or a regular polygon in $\mathbb{R}^2$ with $k$ sides, where $k\geq 4$. Numerical results support this inequality for several classes of convex domains, but the conjecture remains open, particularly the equality characterization.