Weak modularity of the 1-skeleta of A_n buildings

Let a building of type $\widetilde{A}_n$ be a simplicial complex built from Coxeter complexes of type $\widetilde{A}_n$, and let its 1-skeleton be the graph whose vertices and edges are those of the building. A graph is weakly modular if it satisfies the triangle and quadrangle conditions. Weak modularity conjecture for $\widetilde{A}_n$ buildings. The 1-skeleta of buildings of type $\widetilde{A}_n$ are weakly modular. The result is proved in the source for buildings of type $\widetilde{A}_3$; the conjecture asserts that the same conclusion holds for every $n$, although it is unclear whether the given proof generalizes or whether different techniques are required.