On the Chow groups of certain geometrically rational 5-folds

We give an explicit regular model for the quadric fibration studied in [8]. As an application we show that this construction furnishes a counterexample for the integral Tate conjecture in any odd characteristic for some sufficiently large finite field. We study the étale cohomology of this regular model, and as a consequence we derive that these counterexamples are not torsion.