Wavelet completeness conjecture for locally compact groups

Let $G$ be a locally compact group, let $\widehat{G}_{s}$ denote the equivalence classes of square integrable representations of $G$, and for each square integrable representation $\pi:G\to\mathcal{U}(\mathcal{H}_{\pi})$ let $\mathcal{A}_{\pi}$ denote the equivalence classes of admissible vectors in $\mathcal{H}_{\pi}$ modulo rotations by elements of $\mathbb{T}$. For $g\in\mathcal{A}_{\pi}$ and $f\in\mathcal{H}_{\pi}$, write $\mathcal{W}_{g}f$ for the corresponding wavelet transform.

Wavelet completeness conjecture. Characterize the locally compact groups $G$ satisfying

$$\overline{\bigoplus_{\pi\in\widehat{G}_{s}}\underset{g\in\mathcal{A}_{\pi}}{\operatorname{span}}\big\{\mathcal{W}_{g}f:\ f\in\mathcal{H}_{\pi}\big\}}=L^{2}(G).$$

This asks when wavelet spaces associated with all square integrable representations collectively have dense span in $L^{2}(G)$. The source gives the characterization as an open problem; examples show that varying the representation can restore completeness even when varying admissible vectors within one representation does not.