Type-II spectral decomposition conjecture for standard quotients

Let $(G,H,L)$ be a properly transitive triple. Write $\widehat G_H$ for the relevant $H$-cospherical $G$-representations, let $\widehat{\mathbf D(G/H)}$ denote the joint spectrum of $\mathbf D(G/H)$, and let $V_\pi$ be a representation of $L$.

Type-II spectral decomposition conjecture. There exist a topological space $\widehat G_H^L\supset\widehat G_H$, a continuous finite-to-one map

$$p:\widehat G_H^L\longrightarrow \widehat{\mathbf D(G/H)},$$

measurable Hilbert-space fibers $[\operatorname{Hom}_L(W_{\rho,\infty},V_{\pi,\infty})\otimes(W_{\rho,-\infty})^H]_0$, and a $\sigma$-finite Borel measure $\mu_\pi$ such that, for every $\pi\in\widehat L$ with $V_\pi^{L\cap H}\ne\{0\}$, the natural $\mathbf D(G/H)$-equivariant maps induce a unitary equivalence

$$V_\pi^{L\cap H}\cong\int^{\oplus}_{\widehat G_H^L}[\operatorname{Hom}_L(W_{\rho,\infty},V_{\pi,\infty})\otimes(W_{\rho,-\infty})^H]_0\,d\mu_\pi(\rho).$$

This is intended to supply the missing spectral decomposition for standard quotients associated with Type-II triples, independently of the lattice $\Gamma$. The source presents it as a conjecture and gives no resolution status.