Proper locally trivial bundle is injective on Chow groups If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to be injective. 
Is this true and why?

(I keep thinking this should be able to be proven using rigidity lemma and/or Noetherian induction, but I think I'm just being slow.)
 A: First for a positive result.This is true if $P$ is cellular (or more generally when the motive of $P$ splits into a sum of Tate motives), $X$ is smooth, and when the domain has degree larger than the dimension of $Z=Y\setminus U$.
Proof. The claim is true when $Z$ is empty (arbitrary degree):
Since $P$ is cellular, the motive of $U\times P$ decomposes into a product of the shifts of the motive of $U$, and the pullback of $U\times P\rightarrow U$ is the inclusion in appropriate codimension.
The claim is true in general:
From the localization sequences (with commuting squares and exact rows) $$\begin{matrix} \mathrm{CH}_i(Z)&\rightarrow& \mathrm{CH}_i(Y)&\rightarrow& \mathrm{CH}_i(U)&\rightarrow & 0 \\ \downarrow && \downarrow && \downarrow\\  \mathrm{CH}_{i+d}(X\times_Y Z) &\rightarrow &\mathrm{CH}_{i+d}(X)& \rightarrow & \mathrm{CH}_{i+d}(U\times P)&\rightarrow & 0\end{matrix}$$ one can check (by a diagram chase), for $i>\dim(Z)$ the map $\mathrm{CH}_i(Y)\rightarrow \mathrm{CH}_{i+d}(X)$ is an injection. $\square$
A counterexample is explained in the comments. E.g., one can take a Severi-Brauer variety for $Y$ satisfying the property that it's Chow group has torsion. Then a Galois splitting field for $Y$, say $L$, gives a variety $X=Y_L$ such that $\mathrm{CH}(Y)\rightarrow \mathrm{CH}(Y_L)$ is not injective.
