Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs

Let σ0,…,σD denote a pseudo cosine sequence of Γ with σ1∉{−1,1} and let E denote an associated pseudo primitive idempotent. The following are equivalent: (i) the entrywise product of E with itself is a linear combination of the all-ones matrix and a pseudo primitive idempotent of Γ; (ii) there exists a scalar β such that σi−1−βσi+σi+1=0 for 1≤i≤D−1. Moreover, Γ has such a pseudo cosine sequence and pseudo primitive idempotent if and only if Γ is almost 2-homogeneous with c2≥2.