The singularity conjecture for stationary measures on the Furstenberg boundary

Let $\operatorname{\mathsf{G}}$ be a connected semisimple Lie group without compact factors and with finite center. Suppose $\Gamma<\operatorname{\mathsf{G}}$ is a Zariski dense discrete subgroup, and let $\mathsf{m}$ be a probability measure on $\Gamma$ whose support generates $\Gamma$ as a group. Let $\operatorname{\mathcal{F}}=\operatorname{\mathsf{G}}/\operatorname{\mathsf{P}}$ be the Furstenberg boundary, where $\operatorname{\mathsf{P}}$ is a minimal parabolic subgroup of $\operatorname{\mathsf{G}}$, 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 conjecture concerns the relationship between stationary measures arising from random walks on Zariski dense discrete subgroups and the natural Lebesgue measure class on the Furstenberg boundary. The paper proves the assertion under additional hypotheses, including when the ambient semisimple Lie group has property (T), and in rank one when the measure has finite first moment; the general finite-support case stated here remains unresolved based on the supplied context.