Contracting isometry conjecture for groups acting on non-affine buildings

Let $(W,S)$ be an irreducible non-spherical and non-affine Coxeter system with $S$ finite, let $X$ be a building of type $(W,S)$, and let $G$ be a group acting on $X$ by automorphisms without fixing any point at infinity in the ${\rm CAT(0)}$ realization of $X$. Contracting isometry conjecture. Then $G$ either stabilizes a proper residue or contains a contracting isometry. The conjecture proposes a weaker geometric hypothesis than the theorem used in the paper to produce quasi-morphisms, and would clarify when group actions on buildings yield contracting dynamics. Its resolution would also have consequences for rigidity questions concerning higher-rank lattices, via the Burger–Monod vanishing theorem and the Bestvina–Fujiwara construction.