Injectivity of pushforward of rational Chow groups

I’d like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which is the intersection of two hypersurface sections in $X$. Is the pushforward $CH_i(Z)otimes mathbb{Q}rightarrow CH_i(X)otimes mathbb{Q}$ injective?