  We consider Calabi-Yau $n$-folds $X$ arising from certain hyperplane
arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with
motive of abelian type, we show that the subring of the Chow ring of $X$
generated by divisors, Chern classes and intersections of subvarieties of
positive codimension injects into cohomology. We also prove Voisin's conjecture
for $X$, and Voevodsky's smash-nilpotence conjecture for odd-dimensional $X$.
