The closedness conjecture for spectral orbits

Let $G$ be a real reductive group, let $P=LU$ be a parabolic subgroup with abelian unipotent radical $U$, and let $\pi$ be a Casselman–Wallach representation of $G$. Define the spectral orbits of $\pi$ to be the $L$-orbits in $\widehat{U}$ containing some $\phi$ such that

$$\pi / \langle u\cdot v-\phi(u)v\mid u\in U,\,v\in\pi\rangle\neq 0.$$

Let $\mathrm{SO}_U(\pi)$ denote the union of these spectral orbits.

Closedness conjecture for spectral orbits. The subset $\mathrm{SO}_U(\pi)$ is closed in $\widehat{U}$.

The conjecture concerns the relationship between coarse spectral filtrations and nilpotent invariants of representations. Its resolution is not supplied in the source.