Serre's version of the Centre Conjecture for spherical buildings

Let $G$ be a connected group, let $X=X(G,k)$ be its spherical Tits building over $k$, and identify $X$ with its geometric realization. A subcomplex $Y$ of $X$ is convex if, whenever two points of $Y$ are not opposite in $X$, $Y$ contains the unique geodesic joining them; it is contractible if it has the homotopy type of a point. Serre's Centre Conjecture. If $Y$ is a convex and contractible subcomplex of $X$, then there is a point $y\in Y$ fixed by every automorphism of $X$ that stabilizes $Y$. This asserts the existence of a canonical centre for every convex contractible subcomplex; it was proved by M. Mühlherr and J. Tits for spherical buildings of classical type, while the general version stated here is not resolved by the supplied context.