Torsion nilpotence conjecture for algebraic correspondences

Let $k$ be a field, let $X$ be a smooth projective variety of dimension $d$ over $k$, and let $n$ be relatively prime to $\operatorname{char}k$. Consider the torsion cycle class map

$$CH^d(X\times X)[n]\longrightarrow H^{2d}_{\operatorname{\acute et}}(X_{\overline{k}}\times X_{\overline{k}},{\mathbb Z}/n(d)).$$

A correspondence $\Gamma\in CH^d(X\times X)[n]$ is cohomologically trivial when it lies in the kernel of this map.

Torsion nilpotence conjecture. Every cohomologically trivial $n$-torsion correspondence $\Gamma$ is nilpotent.

This is presented as the torsion version of the homological nilpotence conjecture. The paper proves it for powers $n=\ell^r$ for all but finitely many primes $\ell$ when $X$ is a surface over a perfect field, while the general question remains open in the supplied text.