Real Regulators for Products of Elliptic Curves

Assuming the Kunneth decomposition of the Chow groups of products of general Kummer surfaces, we prove that the Hodge-$\mathcal{D}$-conjecture fails for the real regulator $r_{k,1}$ on a product of $n$ general elliptic curves for $2n\ge 3k-1\ge 8$.