Discrete spectrum conjecture for quantum layers over asymptotically flat surfaces

Let $\varSigma$ be a complete, non-compact surface isometrically immersed in $\mathbb{R}^3$ and asymptotically flat. Let $\Omega$ be the quantum layer built over $\varSigma$, and let $K$ denote its Gaussian curvature.

Quantum-layer discrete-spectrum conjecture. If $\varSigma$ is not the plane and $K$ is integrable on $\varSigma$, then the Dirichlet Laplacian on $\Omega$ has non-empty discrete spectrum.

This asks whether every non-planar asymptotically flat surface with integrable Gaussian curvature supports a bound state in its quantum layer. The statement is motivated by results under stronger assumptions, including integrability and non-positivity of the total Gaussian curvature; the general claim remains unresolved in the supplied source.