Charmenability of twin building lattices

Let $Λ= \mathcal G_A (\mathbb F_q)$ be a minimal simply connected split Kac-Moody group of compact-hyperbolic type, and set $Γ=Λ/Z(Λ)$. We prove that $Γ$ is charfinite when $q\geq 11$, or when its Weyl group is $3$-spherical and $q\geq 4$. Here are a few consequences: the only characters of $Γ$ are