Generalized singularity conjecture for coarse Iwasawa Patterson–Sullivan measures

Let $\operatorname{\mathsf{G}}$ be a connected semisimple Lie group as in the surrounding setup, let $\operatorname{\mathcal{F}}=\operatorname{\mathcal{F}}_{\Delta}$ be its Furstenberg boundary, and let $\nu$ be the $\mathsf{m}$-stationary measure associated to a probability measure $\mathsf{m}$ on a Zariski dense discrete subgroup $\Gamma<\operatorname{\mathsf{G}}$ whose support generates $\Gamma$ as a semigroup. A coarse Iwasawa Patterson–Sullivan measure is a coarse $\phi$-Patterson–Sullivan measure on a partial flag manifold arising from the partial Iwasawa cocycle.

Generalized singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to every coarse Iwasawa Patterson–Sullivan measure on $\operatorname{\mathcal{F}}$.

This generalizes the preceding singularity conjecture from the Lebesgue measure class to coarse Iwasawa Patterson–Sullivan measures. The source motivates the claim by the existence of an Iwasawa Patterson–Sullivan measure in the Lebesgue measure class, but does not state a resolution.