Strong twisted Hodge D conjecture

Let $X$ be a smooth projective variety, and let $\underline{CH}^{k}(X,m)$ be the relevant higher Chow group, $\underline{r}_{k,m}$ its regulator, and $D(k,m)$ the indicated denominator subgroup. Strong twisted Hodge $\mathcal D$ conjecture. The morphism

$$\underline{r}_{k,m} \otimes \mathbb{R}: \underline{CH}^{k}(X,m) \otimes \mathbb{R} \rightarrow \frac{\big\{H^{k-1,k-m}(X)\oplus H^{k-m,k-1}(X)\big\} \cap H^{2k-m-1}(X,\mathbb{R})}{D(k,m)}$$

is surjective. This is the strong form of the twisted Hodge $\mathcal D$ conjecture introduced for studying regulators of twisted cycles; the supplied text does not establish its general status.