Strict tessellation characterization of trigonometric Laplace eigenfunctions

Let $\Omega$ be a domain in $\mathbb{R}^n$. A complete set of trigonometric eigenfunctions means a complete set of eigenfunctions of the Laplace eigenvalue equation on $\Omega$ with Dirichlet boundary condition, each of trigonometric form. A polytope strictly tessellates $\mathbb{R}^n$ when its reflected copies form the relevant tessellation; an alcove is such a polytope associated with a crystallographic reflection group.

Strict tessellation conjecture. $\Omega$ has a complete set of trigonometric eigenfunctions for the Laplace eigenvalue equation with the Dirichlet boundary condition if and only if $\Omega$ is a polytope which strictly tessellates $\mathbb{R}^n$. Equivalently, this holds if and only if $\Omega$ is an alcove.

The supplied text presents this as a conjecture related to, but independent of, Fuglede's conjecture. Its resolution is not established by the supplied material.