Tits' center conjecture for spherical buildings

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