Existence of a ground state for quantum layers with integrable Gaussian curvature

Let $p:\varSigma\to\mathbb R^3$ be an embedded, orientable, complete, non-compact surface that is not totally geodesic and whose second fundamental form tends to zero at infinity. For a positive number $a$, let $\Omega=\varSigma\times[-a,a]$ with the quantum-layer metric induced by

$$\tilde p(x,t)=p(x)+t\vec N_x,$$

where $\vec N$ is the unit normal vector to $\varSigma$, and let $\Delta=\Delta_\Omega$ be the Dirichlet Laplacian. Ground-state existence conjecture. If

$$\int_\varSigma |K|\,d\varSigma<+\infty,$$

then the ground state of $\Delta$ exists. The claim concerns the existence of a bound state for the Dirichlet Laplacian of an asymptotically flat quantum layer under an integrability assumption on the Gaussian curvature; the supplied text gives no evidence of resolution, so its status is open.