Let $A$ be a $g$-dimensional abelian variety, $H$ an ample divisor, $D\in Pic^0(A)$, and $0\leq k\leq g$. Then if $D^k\neq 0$, is it true that $H^{g-k}\cdot D^k\neq 0$? We are working in the Chow group of $A$ (cycles modulo rational equivalence) with rational coefficients, but I'm interested in the case with integer coefficients as well.