The Furstenberg-boundary criterion for simplicity of the reduced group C*-algebra

Let $G$ be a second countable locally compact group. Furstenberg-boundary simplicity conjecture. If some point of the Furstenberg boundary of $G$ has a trivial stabilizer, then

$$C^*_r(G)\text{ is simple}.$$

This extends the corresponding result known for discrete groups to non-discrete locally compact groups; the source states that the general case is currently unknown.