Beilinson's Hodge-$\mathcal D$-Conjecture

Let $X$ be a smooth variety over $\bar{\mathbb{Q}}$. For integers $p$ and $n$, let $CH^p(X,n)$ denote the higher Chow group, let $H^{2p-n}_{\mathcal{D}}(X,\mathbb{R}(p))$ denote Deligne cohomology with real coefficients, and define the real regulator after extension of scalars by

$$r_{\mathcal{D},\mathbb{R}}^{p,n}\coloneqq r_{\mathcal{D}}^{p,n}\otimes\mathbb{R}\colon CH^p(X,n)\otimes\mathbb{R}\to H^{2p-n}_{\mathcal{D}}(X,\mathbb{R}(p)).$$

Hodge-$\mathcal D$-Conjecture. The map $r_{\mathcal{D},\mathbb{R}}^{p,n}$ is surjective. This is a higher-dimensional and higher-cycle analogue of the Hodge conjecture, formulated through the real regulator because the generalized Hodge classes vanish for smooth projective varieties. Its status is not resolved in the supplied source.