A remark on a $3$-fold constructed by Colliot-Thel`ene and Voisin

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-Thel`ene and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.