Let $A$ be a $g$-dimensional, complex abelian variety, let $H$ be an ample divisor, let $D\in Pic^0(A)$, and let $0\leq k\leq g$.

Question 1: Does $D^k\neq 0\in CH^k(A,\mathbb{Q})$ imply that $H^{g-k}\cdot D^k\neq 0\in CH^g(A,\mathbb{Q})$?

(As Jason points out, it is necessary to work with $\mathbb{Q}$-coefficients.)

Question 2: Assuming the answer to Question 1 is "yes", is there a simple proof of this fact?

Primarily, we are interested in abelian varieties over $\mathbb{C}$. (As ulrich points out, Kunnemann answered Question 1 affirmatively for abelian varieties over finite fields.)