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Let $E$ be the Fermat cubic curve over $\overline{\mathbb{Q}}$. 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 primenumbers $\ell> 5$. This gives the first example of a smooth projective variety$X$ over $\overline{\mathbb{Q}}$ such that $CH^2(X)/\ell$ is infinite for allbut at most finitely many primes $\ell$. A key tool is a recent theorem ofFarb--Kisin--Wolfson, whose proof uses the prismatic cohomology ofBhatt--Scholze.

Varieties over $\overline{\mathbb{Q}}$ with infinite Chow groups modulo almost all primes