This is a report on joint work with A. Rosenschon. We show that on such a
3-fold, for all but a finite number of positive integers $n$, the Chow group
of curves with mod $n$ coefficients is not finitely generated. This is done in
two steps: first we use a variant of the technique of Bloch and Esnault to
show that the Ceresa cycle is not $n$-divisible for almost all $n$. Then we
use modular correspondences, following Nori, to show infinite generation.