I am in a situation where I need a result of the following form. Suppose $X$ is a smooth $k$-variety, $U$ is a dense open subvariety with complement $Z$ a smooth divisor. Let $\pi^X:X\rightarrow\text{Spec }k$ and $\pi^U:U\rightarrow\text{Spec }k$ be the structure morphisms. Let us work with Voevodsky motives $\text{DM}(k)$. Is it true that if $\pi^X_!\mathbf{1}_X(-n)$ is effective for some positive integer $n$, then so is $\pi^U_!\mathbf{1}_U(-n)$?