  A classical question asks whether the Abel-Jacobi map is universal among all
regular homomorphisms. In this paper, we prove that we can construct a $4$-fold
which gives the negative answer in codimension $3$ if the generalized Bloch
conjecture holds for a $3$-fold constructed by Colliot-Th\'el\`ene and Voisin
in the context of the study of the defect of the integral Hodge conjecture in
degree $4$.
