The boundary-representation C*-simplicity conjecture

Let $G$ be a second countable locally compact group and let $\partial_F G$ be its Furstenberg boundary. For $z\in\partial_F G$, assume that $z$ is a continuity point of the stabilizer map

$$\partial_F G\to \operatorname{Sub}(G).$$

Boundary-representation C-simplicity conjecture.* The $C^*$-algebra associated with the quasi-regular representation on $G/G_z$ is simple:

$$C^*_{\lambda_{G/G_z}}(G)\text{ is simple}.$$

Equivalently, in the source's URS notation, $C^*(\mathcal A)$ should be simple for the stabilizer URS $\mathcal A$ of the action on the Furstenberg boundary. This is presented as a strengthening of the preceding conjecture and remains open in the source.