Radius-gap conjecture for convex subsets of spherical buildings

Let $B$ be a spherical building and let $C\subseteq B$ be a closed convex subset. Write $rad_C(C)$ for the intrinsic radius of $C$, namely the infimum of the radii of balls centered in $C$ and containing $C$. Radius-gap conjecture. Either $C$ is a subbuilding, or

$$rad_C(C)\leq\frac{\pi}{2}.$$

This asks whether there is a gap between the intrinsic-radius-$\pi$ case, which forces $C$ to be a building, and the intrinsic-radius-at-most-$\frac{\pi}{2}$ case, which gives the fixed-point property. The source presents this as a more general gap phenomenon related to the fixed-point conjecture.