Strongly Closed Subgraphs in a Distance-Regular Graph with c2 > 1

Let Γ be a distance-regular graph of diameter d ≥ 3 with c2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions: (SC)m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them.(BB)m : Let (x, y, z) be a triple of vertices with ∂Γ (x, y) = 1 and ∂Γ (x, z) = ∂Γ (y, z)  =  m. Then B(x, z) = B(y, z).(CA)m : Let (x, y, z) be a triple of vertices with ∂Γ (x, y) = 2, ∂Γ (x, z) = ∂Γ (y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z).