Langford–Laugesen two-disk Neumann eigenvalue conjecture

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain, let $\omega\in C^2(\Omega)\cap C(\overline{\Omega})$ be positive on $\overline{\Omega}$, and equip $\Omega$ with the metric $g=\omega|dz|^2$. Suppose that the Gaussian curvature satisfies $K_g\le K$, let $A=\int_\Omega\omega\,dx>0$, and assume $KA<4\pi$ whenever $K>0$. If the Neumann eigenvalues are enumerated as $0=\lambda_0(\Omega,g)<\lambda_1(\Omega,g)\le\lambda_2(\Omega,g)\le\cdots$, then the conjecture asserts that $\lambda_2(\Omega,g)<\lambda_1\!\left(D_K(A/2)\right)$, where $D_K(A/2)$ is the geodesic disk of area $A/2$ in the constant-curvature-$K$ model. The bound is sharp in the class of connected disk-type membranes, with extremizing sequences degenerating to the disjoint union of two copies of $D_K(A/2)$.