Real regulator conjecture for products of very general elliptic curves

Let $E_1,E_2,\ldots,E_n$ be very general complex elliptic curves and let

$$X=E_1\times E_2\times\cdots\times E_n.$$

For $1\leq m\leq n$, define

$$T_m\bigl(H^{2n-2k+2}(X)\bigr)=\sum_{\substack{|I|=m\\ I\subset\{1,2,\ldots,n\}}}\bigotimes_{i\in I}H^1(E_i)\otimes H^{2n-2k+2-m}\left(\prod_{j\notin I}E_j\right).$$

The real regulator is the map $r_{k,1}$ on $\operatorname{CH}_{\mathbb R}^k(X,1)$. Real regulator conjecture. The map $r_{k,1}$ is surjective for $k=2$ and is trivial, in the sense that its image is orthogonal to one of the subspaces $T_m(H^{2n-2k+2}(X))$, for all $2n\geq 3k-1\geq 8$. More precisely, for all $3\leq k\leq 2r+1$, one expects

$$r_{k,1}\bigl(\operatorname{CH}_{\mathbb R}^k(X,1)\bigr)\subset T_{2r+2}\bigl(H^{2n-2k+2}(X)\bigr)^\perp.$$

For example, when $(k,r,n)=(3,1,4)$,

$$r_{3,1}\bigl(\operatorname{CH}_{\mathbb R}^3(X,1)\bigr)\subset T_4\bigl(H^4(X)\bigr)^\perp=\bigl(H^1(E_1)\otimes H^1(E_2)\otimes H^1(E_3)\otimes H^1(E_4)\bigr)^\perp.$$

This refines the expected behavior of the Hodge-$\mathscr D$-conjecture for real regulators: surjectivity is known in some low-codimension cases, while the asserted orthogonality and failure of surjectivity in the stated range remain conjectural.