Maximal left ideal classification conjecture for Hermitian locally compact groups

Let $G$ be a Hermitian locally compact group. For a representation $pi$ of $G$ and a unit vector $xi$ in its representation space $\mathcal{H}_pi$, define

$$\mathcal{J}_{\pi,\xi}=\{\mu\in M(G):\pi(\mu)\xi=0\}$$

and, for $f\in L^1(G)$, define

$$\mathcal{I}_{\pi,\xi}=\{f\in L^1(G):\pi(f)\xi=0\}.$$

A representation is vanishing at infinity when its matrix coefficients vanish at infinity. Maximal left ideal classification conjecture. The weak*-closed maximal left ideals of $M(G)$ are given exactly by $\mathcal{J}_{\pi,\xi}$, where $\pi$ is an irreducible representation vanishing at infinity, and $\xi$ is a unit vector in $\mathcal{H}_\pi$ with the property that $\mathcal{I}_{\pi,\xi}$ is a maximal modular left ideal of $L^1(G)$. This conjecture proposes an analogue of the classification of maximal modular left ideals in $L^1(G)$ and of Barnes' theorem for integrable representations; the paper proves it for a large class of Hermitian locally compact groups, but it remains open in the stated generality.