Marquis's fixed-point conjecture for locally elliptic actions on buildings

Let $G$ be a group acting by type-preserving simplicial isometries on a building $\Delta$. Let $X$ be the Davis realization of $\Delta$, and suppose that the $G$-action on $X$ is locally elliptic.

Marquis's conjecture. The group $G$ has a global fixed point in $X\cup\partial X$.

This conjecture is confirmed in the special case of right-angled buildings, because their Davis realizations are finite-dimensional $\mathrm{CAT}(0)$ cube complexes and Corollary B applies. The general case is not resolved by the supplied text.