  A method of Gabber (2002) produces unramified cohomology classes in the
products of certain varieties with an elliptic curve. The connection between
third unramified cohomology and integral Hodge conjecture for codimension 2
cycles (2012, which builds upon results from algebraic K-Theory, then gives
many examples of such a product for which this conjecture fails. The special
case of the product with an Enriques surface was established by Benoist and
Ottem (2018).
