A distance-regular graph with bipartite geodetically closed subgraphs

Let Γ be a distance-regular graph of diameter d, and t be an integer with 2≤t≤d−1 such that at−1=0. For any pair (u,v) of vertices, let Π(u,v) be the subgraph induced by the vertices lying on shortest paths between u and v. We prove that if Π(u,v) is a bipartite geodetically closed subgraph for some pair (u,v) of vertices at distance t, then Π(x,y) is a bipartite geodetically closed subgraph for any pair (x,y) of vertices at distance less than or equal to t. In particular, Π(x,y) is either a path, an ordinary polygon, a hyper cube or a projective incidence graph.