Singularity conjecture for stationary measures on Furstenberg boundaries

Let $\operatorname{\mathsf{G}}$ be a connected semisimple Lie group without compact factors and with finite center, let $\Gamma<\operatorname{\mathsf{G}}$ be a Zariski dense discrete subgroup, and let $\mathsf{m}$ be a probability measure on $\Gamma$ whose support generates $\Gamma$ as a semigroup. Let $\operatorname{\mathcal{F}}$ be the Furstenberg boundary, namely the flag manifold associated to a minimal parabolic subgroup, and let $\nu$ be the unique $\mathsf{m}$-stationary measure on $\operatorname{\mathcal{F}}$.

Singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to the Lebesgue measure class on $\operatorname{\mathcal{F}}$.

This is presented as a well-known conjecture concerning Furstenberg measures, with Kaimanovich–Le Prince cited for comparison. The source does not specify whether it is open or resolved.