Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map from Pic$(X) \otimes \mathbb Q_l$ to the Galois invariants of the second etale cohomology group of $X$ is surjective (and thus bijective).

This conjecture is known if X is dominated by a product of curves.

Is it known if X is an etale cover of a product of curves? (We can restrict our attention to the case that X is of positive Kodaira dimension.)

Etale covers of a product of curves are not necessarily a product of curves. We would be done if etale covers of a product of curves are dominated by a product of curves. Is that the case?