# 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?