Cycles of small codimension on a simple $2p$- or $4p$-dimensional Abelian variety

Let $J$ be a simple $2p$- or $4p$-dimensional Abelian variety over the field of complex numbers, where $p\ne 5$ is a prime number. Assume that one of the following conditions holds:
1) $\operatorname{Cent\,End}^0(J)$ is a totally real field of degree 1, 2 or 4 over $\mathbb Q$;
2) $J$ is a simple $2p$-dimensional Abelian variety of CM-type $(K,\Phi)$ such that $K/\mathbb Q$ is a normal extension;
3) $J$ is a simple $2p$-dimensional Abelian variety such that $\operatorname{End}^0(J)$ is an imaginary quadratic extension of $\mathbb Q$.
Then for every positive integer $r<p$ the $\mathbb Q$-space $H^{2r}(J,\mathbb Q)\cap H^{r,r}$ is spanned by cohomology classes of intersections of divisors.