  In the present paper, we provide general bounds for the torsion in the
codimension 2 Chow groups of the products of projective homogeneous surfaces.
In particular, we determine the torsion for the product of four Pfister quadric
surfaces and the maximal torsion for the product of three Severi-Brauer
surfaces. We also find an upper bound for the torsion of the product of three
quadric surfaces with the same discriminant.
