Fixed-point conjecture for convex subsets of spherical buildings

Let $B$ be a spherical building and let $C\subseteq B$ be a closed convex subset. Fixed-point conjecture. Either $C$ is a subbuilding, or the action

$$Isom(C)\curvearrowright C$$

has a fixed point. This generalizes the Center Conjecture from convex subcomplexes to arbitrary closed convex subsets and replaces the stabilizer action by the full isometry group. The source reports that the conjecture was known when $\dim(C)\leq 2$; the paper studies the remaining cases.