Spectral bipartiteness in generalized odd graphs of diameter three

For a graph $G$ of order $n$, put $σ(G)=(λ_1(G)+λ_n(G))/n$. We determine the first three largest values of this invariant among nonbipartite distance-regular graphs of diameter three and odd girth at least seven. The unique maximizer is the folded $7$-cube, with value $1/32$; the unique second maxim