Kuykian fields

The Kuykian conjecture for a Hilbertian field K says that if A/K is an abelian variety, then every intermediate field of K(A(tor))/K is Hilbertian. We prove the Kuykian conjecture in the following cases: (a) K is finitely generated (over its prime field). (b) K = F-s[sigma] for almost all sigma is an element of Gal(K)(e), where F is a finitely generated field. (c) K = F-ins, where F is the quotient field of a complete local domain of dimension at least 2.