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

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