On the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field

Let $X$ be the product of a smooth projective curve $C$ and a smooth projective surface $S$ over a field $K$. Assume the Chow group of zero-cycles on $S$ is just $Z$ over any algebraically closed field extension of $F$ (example : Enriques surface). For $K$ the complex field, one may give counterexamples to the integral Hodge conjecture for 1-cycles (Benoist-Ottem) on $X$ and this may be understood from the point of view of unramified cohomology. For $K$ a finite field, in joint work with Federico Scavia (UBC, Vancouver) we give a simple condition on $C$ and $S$ which ensures that the integral Tate conjecture holds for 1-cycles on $X$. An equivalent formulation is a vanishing result for unramified cohomology of degree 3.