Completeness of compressed plane waves

Let $\mathbf{w}=(w_1,\cdots,w_d)\in\mathbb{R}_+^d$ define the lattice $\Gamma_{\mathbf{w}}=\{\mathbf{j}\mathbf{w}:\mathbf{j}\in\mathbb{Z}^d\}$. Let $\{\psi^n\}_{n=1}^{\infty}$ be the localized orthonormal modes recursively generated by the displayed variational problems, and define

$$\mathrm{b}^n_{\mathbf{j}}(\mathbf{x})=\psi^n(\mathbf{x}-\mathbf{j}\mathbf{w}),\qquad n\geq1,\quad \mathbf{j}\in\mathbb{Z}^d.$$

Completeness of CPWs. There exists a constant $\mu_0$ such that the set $\{\mathrm{b}^n_{\mathbf{j}}\}_{n,\mathbf{j}}$ generated from the recursive variational definitions is complete for every $\mu\geq\mu_0$. The conjecture concerns whether the compressed plane waves form a complete orthonormal system; the passage explicitly says that this will be studied in future work and provides no resolution.