Rost nilpotence principle for smooth projective schemes

Let $X$ be a smooth projective equi-dimensional scheme over a field $k$. For a field extension $E/k$, write $[?]$ for the base change of a correspondence to $X_E$. A correspondence is an element of

$$\operatorname{End}_k(X)=\operatorname{CH}^{{\mathrm{dim}}(X)}(X\times X).$$

Rost nilpotence principle. For every $\Gamma\in \operatorname{End}_k(X)$ such that $\Gamma_E=0$ for some field extension $E/k$, $\Gamma$ is nilpotent as a correspondence.

The principle asserts that a correspondence that becomes zero after scalar extension already acts nilpotently over the original field. The supplied text states the principle but gives no information about whether it is proved or remains open.