In Klyachko's paper "Equivariant vector bundles on toral varieties" I saw a statement about the Tits building of $\operatorname{GL}(n, \mathbb{C})$. I was wondering if this statement/property holds for more general buildings. Klyachko calls it "an analogue of Helly's theorem" on convex sets in Euclidean space.

Question: Let $S$ be a finite collection of simplexes in a (spherical) building $B$ of rank $n-1$ (I only care for the case when $B$ is the Tits building of a reductive algebraic group over $\mathbb{C}$). Suppose any $n+1$ simplexes in $S$ lie in a common apartment. Is it true that all the simplexes in $S$ lie in a common apartment?