Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension.

For any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(X_K)$), does there exist an algebraic cycle $\xi\in Z^1(X)$ (ie. defined over $k$) such that $\xi_K-\eta$ is effective?

Fix a collection $\{Y_1,\ldots, Y_n\}$ of $k$-subvarieties of $X$, and further assume each component of $\eta$ intersects each of the $(Y_i)_K$'s in prescribed fixed codimension $p$. Does there exist $\xi\in Z^1(X)$ such that each component of $\xi$ intersects each of the $Y_i$'s in codimension $p$, and $\xi_K-\eta$ is effective?