Denote $CHM(F)$ to be the category of Chow motives over a field $F$.
Let's consider an algebraic exension $E/F$, then there is a natural extension of scalars functor $CHM(F) \to CHM(E)$.
I was wondering if this functor is conservative, i.e. if a morphism $f: M \to N$ becomes an isomorphism after a field extension, does it imply $f$ is an isomorphism itself?
A related question is: if a motive $M$ becomes zero after a field extension, does it imply that $M = 0$? I believe this question is weaker, than that of being conservative.
Merkurjev-Gille-Chernousov (Corollary 8.4) prove this for motives of homogenous spaces for algebraic group actions (so-called Rost Nilpotence theorem, since it was originally prove by Rost for quadrics).
Do people believe that this holds in general? Is it related to some standard motivic conjectures?