  Let $E$ be the Fermat cubic curve over $\bar{\mathbb{Q}}$. In 2002, Schoen
proved that the group $CH^2(E^3)/\ell$ is infinite for all primes $\ell\equiv
1\pmod 3$. We show that $CH^2(E^3)/\ell$ is infinite for all prime numbers
$\ell> 5$. This gives the first example of a smooth projective variety $X$ over
$\bar{\mathbb{Q}}$ such that $CH^2(X)/\ell$ is infinite for all but at most
finitely many primes $\ell$. A key tool is a recent theorem of
Farb--Kisin--Wolfson, whose proof uses the prismatic cohomology of
Bhatt--Scholze.
