Tits' centre conjecture for spherical buildings

Let $X$ be a spherical building, and let $Y$ be a convex contractible simplicial subcomplex of $X$. Let $H$ be a subgroup of the automorphism group of $X$ that stabilizes $Y$. Tits' centre conjecture. There exists a simplex of $Y$ fixed by $H$. This conjecture concerns fixed-point phenomena for group actions on convex subcomplexes of spherical buildings. It was proved by case-by-case analyses by Tits, Mühlherr, Leeb, and Ramos-Cuevas.