On bipartite distance-regular graphs with exactly one non-thin T-module with endpoint two

Fix x∈X and assume that Γ has, up to isomorphism, exactly one irreducible T-module W with endpoint 2, and that W is non-thin with dim(E2∗W)=1, dim(ED−1∗W)≤1 and dim(Ei∗W)≤2 for 3≤i≤D. We prove that for 2≤i≤D, there exist complex scalars αi,βi such that |Γi−1(x)∩Γi−1(y)∩Γ1(z)|=αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)| for all y∈Γ2(x) and z∈Γi(x)∩Γi(y). Furthermore, we prove Δ2=0 and either D=5 or c2∈{1,2}. We show there exist integers 3≤f≤ℓ≤D−2 such that dim(Ei∗W)=2 if and only if f≤i≤ℓ.