Reference for boundary regularity of Neumann eigenfunctions

Question

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with piecewise $C^{\infty}$ boundary. I have seen it implicitly used in several results that Neumann eigenfunctions of the Laplacian on $\Omega$ are $C^{\infty}$ at the smooth points of the boundary, but I cannot find a reference for this anywhere. In the Dirichlet case, the analogous result can be obtained by a minor modification to the proof of Theorem 6 in the textbook by Evans. I would also be happy just to see a reference for the result when the entire boundary is $C^{\infty}$.