Locally elliptic actions on Davis realizations have a global fixed point

Let $G$ be a group acting by type-preserving simplicial isometries on a building $\Delta$. Let $X$ be the Davis realization of $\Delta$, and call the action locally elliptic if every element of $G$ has a fixed point in $X$. Fixed-point conjecture. If the $G$-action on $X$ is locally elliptic, then $G$ has a global fixed point in

$$X\cup\partial X,$$

where $\partial X$ denotes the visual boundary of $X$. This would generalize the fixed-point result for almost connected locally compact groups acting on trees and would imply the preceding conjecture that every almost connected locally compact group has property $\operatorname{(FB)}$; the status of the assertion is not resolved in the supplied text.