A characterization of two classes of locally truncated diagram geometries

Let Γ = (P,L) be a parapolar space which is locally An−1,3(K) for some integer n > 6 and K a field. There exists a class of 2-convex subspaces D, each isomorphic to D5,5(K), such that every symplecton of Γ is contained in a unique element of D. Let Γ = (P,L) be a parapolar space which is locally An−1,4(K) for n = 7 or an integer n ≥ 9 and some field K. Assume that Γ satisfies the extra condition called the Weak Hexagon Axiom. Then there exists a class of 2-convex subspaces D, each isomorphic to D6,6(K), such that every symplecton of Γ is contained in a unique element of D. In both of the above cases Γ is the homomorphic image of a truncated building.