Let $D\subset R^d$ be a bounded Lipschitz domain. We know that the Neumann eigenfunction lies in $C(\overline{D})$ (i.e. continuous up to the boundary). This can be seen from the fact that $\phi_k=e^{\lambda_kt}P_t\phi_k$ where $P_t$ is the semigroup of the reflected Brownian motion.

Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$?

If yes, what's the reason? If not, what is the closure of the linear span with respect to the supremum norm?