Nilpotence conjecture for homologically trivial correspondences of Chow motives

Let $k$ be a field, let $R$ be either ${\mathbb Z}$ or ${\mathbb Q}$, and let ${\mathcal M}_{k,R}$ be the category of Chow motives over $k$ with $R$ coefficients. For a Chow motive $M\in {\mathcal M}_{k,R}$, write $\operatorname{End}_{{\mathcal M}_{k,R}}(M)$ for its endomorphism ring. An endomorphism $\Gamma$ is homologically trivial when $\Gamma\sim_{hom}0$.

Nilpotence conjecture. If $\Gamma\in \operatorname{End}_{{\mathcal M}_{k,R}}(M)$ is homologically trivial, then there exists an integer $n\gg0$ such that

$$\Gamma^n=0\in \operatorname{End}_{{\mathcal M}_{k,R}}(M).$$

This is a nilpotence conjecture for homologically trivial correspondences and concerns the kernel of the cycle class map for Chow motives. The supplied text gives no resolution status for the conjecture.