On the Chow ring of certain rational cohomology tori

Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that $f^*$ induces an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$ and $H^\bullet(X, \mathbb{Q})$ if and only if $f^*$ induces an isomorphism between the Chow rings with rational coefficients $\mathrm{CH}^\bullet(A)_{\mathbb{Q}}$ and $\mathrm{CH}^\bullet(X)_{\mathbb{Q}}$.