The local eigenvalues of a bipartite distance-regular graph

We consider a bipartite distance-regular graph ΓΓ with vertex set XX, diameter D≥4D≥4, and valency k≥3k≥3. For 0≤i≤D0≤i≤D, let Γi(x)Γi(x) denote the set of vertices in XX that are distance ii from vertex xx. We assume there exist scalars r,s,t∈Rr,s,t∈R, not all zero, such that r|Γ1(x)∩Γ1(y)∩Γ2(z)|+s|Γ2(x)∩Γ2(y)∩Γ1(z)|+t=0r|Γ1(x)∩Γ1(y)∩Γ2(z)|+s|Γ2(x)∩Γ2(y)∩Γ1(z)|+t=0 for all x,y,z∈Xx,y,z∈X with path-length distances ∂(x,y)=2,∂(x,z)=3,∂(y,z)=3∂(x,y)=2,∂(x,z)=3,∂(y,z)=3. Fix x∈Xx∈X, and let Γ22 denote the graph with vertex set X̃={y∈X∣∂(x,y)=2} and edge set R̃={yz∣y,z∈X̃,∂(y,z)=2}. We show that the adjacency matrix of the local graph Γ22 has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if ΓΓ is QQ-polynomial.