Singularity of harmonic measures for hyperbolic lattices

Let $Γ<\mathrm{Isom}(\mathbb H^n)$, $n\geq2$, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $Γ$ has hitting measure singular with respect to Lebesgue measure on $\partial\mathbb H