Nonexistence of admissible vectors for monomial representations of unimodular exponential solvable Lie groups

Let $G$ be a unimodular exponential solvable Lie group, and let $\tau$ be a monomial representation of $G$. A vector $\eta$ is admissible if the associated wavelet transform $W_\eta$ is an isometry into $L^2(G)$. Nonexistence conjecture. The representation $\tau$ never has an admissible vector. The preceding proposition characterizes admissibility of closed left invariant subspaces through the multiplicity function of the Plancherel decomposition; in the unimodular case, examples suggest that this multiplicity function is never integrable for monomial representations. The claim is therefore presented as a conjecture for the unimodular case.