Tits' Center Conjecture for convex subcomplexes of spherical buildings

Let $B$ be a spherical building and let $K\subseteq B$ be a convex subcomplex. A center is a point fixed by the automorphisms of $B$ preserving $K$. Tits' Center Conjecture. Either $K$ is a subbuilding, or the action

$$Stab_{Aut(B)}(K)\curvearrowright K$$

has a fixed point. The conjecture was proposed by J. Tits and concerns fixed-point phenomena for convex subcomplexes; the source abstract states that the result is proved for buildings of type $E_7$ or $E_8$, completing the proof for thick spherical buildings together with earlier work.