Pólya's conjecture for Dirichlet and Neumann Laplacian counting functions

Let $\Omega$ be a bounded Euclidean domain, and let its Dirichlet and Neumann Laplacians have eigenvalue counting functions defined by counting eigenvalues below a threshold. The leading term of Weyl's law is the corresponding phase-space volume term.

Pólya's conjecture. The eigenvalue counting function of the Dirichlet Laplacian on $\Omega$ can be estimated from above by the leading term of Weyl's law, while the eigenvalue counting function of the Neumann Laplacian can be estimated from below by the leading term of Weyl's law.

The conjecture is a central problem in spectral geometry, and the versions for both the Dirichlet and Neumann Laplacians remain open in general.