A Friedlander type estimate for Stokes operators

Let $\Omega \subset \mathbb{R}^d$ be a bounded open connected set with Lipschitz boundary. Let $A^N$ and $A^D$ be the Neumann Stokes operator and Dirichlet Stokes operator on $\Omega$, respectively. Further let $\lambda_1^N \leq \lambda_2^N \leq \ldots$ and $\lambda_1^D \leq \lambda_2^D \leq \ldots$ be the eigenvalues of $A^N$ and $A^D$ repeated with multiplicity, respectively. Then $$\lambda_{n+1}^N < \lambda_n^D$$ for all $n \in \mathbb{N}$.