Algebraic cycles and Verra fourfolds

A Verra fourfold is a smooth projective complex variety defined as a double cover of P^2x P^2 branched along a divisor of bidegree (2,2).
These varieties are similar to cubic fourfolds in several ways (Hodge theory, relation to hyperkaehler fourfolds, derived categories).
Inspired by these analogies, I consider the Chow ring of a Verra fourfold. Among other things, I will show that the multiplicative structure of this Chow ring has a curious K3-like property.