Kaimanovich–Masur singularity conjecture for mapping class groups

Let $\Sigma$ be a closed connected orientable surface of genus at least two, let $\Gamma<\operatorname{Mod}(\Sigma)$ be a non-elementary subgroup, and let $\mathsf{m}$ be a probability measure on $\Gamma$ whose support generates $\Gamma$ as a semigroup. Let $\nu$ be the unique $\mathsf{m}$-stationary measure on Thurston's boundary $\mathcal{PMF}$.

Kaimanovich–Masur singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to every Busemann Patterson–Sullivan measure for $\Gamma$.

This conjecture concerns the relationship between random-walk hitting measures and geometric Patterson–Sullivan measures on the Thurston boundary. The source presents it as a conjecture suggested by Kaimanovich–Masur; its resolution status is not specified.