Guivarc'h–Kaimanovich–Ledrappier singularity conjecture for stationary measures

Let $\Gamma<\operatorname{PSL}_2(\mathbb{R})$ be a lattice, and let $m$ be a finitely supported measure whose support generates $\Gamma$. Let the corresponding stationary measure be the probability measure on the circle satisfying the stationarity equation for the action of $\Gamma$ and the measure $m$. Guivarc'h–Kaimanovich–Ledrappier conjecture. The corresponding stationary measure on the circle is singular with respect to Lebesgue measure. The conjecture concerns the regularity dichotomy for stationary measures, and the source attributes it to Y. Guivarc'h, V. Kaimanovich, and F. Ledrappier. It was proven by Y. Guivarc'h and Y. Le Jan for non-cocompact lattices, while the stated general case remains open.