  The Voevodsky nilpotence conjecture was proved by Bernardara-Marcolli-Tabuada
for certain quadric fibrations, intersections of quadrics, linear sections of
Grassmannians, linear sections of determinantal varieties, and Moishezon
manifolds, and later by Ornaghi-Pertusi for certain cubic fourfolds and
Gushel-Mukai fourfolds. In this paper we refine these results by using the
algebraic equivalence relation instead of the nilpotence equivalence relation.
Along the way, we address also certain cases of K\"uchle fourfolds, families of
Sextic del Pezzo surfaces and families of Fano fourfolds of K3 type.
