Sensible Caveman conjecture

Let a $Q$-polynomial association scheme have $3$ or more classes, and suppose that it is primitive, meaning that it has no nontrivial equivalence relation arising from a union of relations. Its splitting field is the field generated by the eigenvalues of the scheme. Sensible Caveman conjecture. For $Q$-polynomial schemes of $3$ or more classes, if the scheme is primitive then its splitting field is rational. The source presents the preceding three-class question as a special case of this conjecture and gives no evidence of a resolution.