Weak Rost Nilpotence Principle

Let $k$ be a perfect field, let $X$ be a smooth projective scheme over $k$, and let $M(X)$ be its Chow motive in the category ${\mathcal M}_{k}$ of Chow motives. For a field extension $E/k$, write $X_E=X\times_k E$, and let $CH_0(X_E)$ denote the Chow group of zero-cycles. Let $\Gamma\in\operatorname{End}_{{\mathcal M}_{k}}(M(X))$. Weak Rost Nilpotence Principle. If there is some field extension $F/k$ for which

$$\Gamma_F=0\in\operatorname{End}_{{\mathcal M}_{F}}(M(X_F)),$$

then, for every field extension $L/k$, the induced endomorphism

$$\Gamma_{L*}:CH_0(X_L)\longrightarrow CH_0(X_L)$$

is nilpotent. The usual Rost Nilpotence Principle is known for quadrics in characteristic different from $2$, surfaces, homogeneous varieties, and certain threefolds, but is open in general; the paper studies whether this weaker assertion suffices to recover the full principle over characteristic-zero fields.