The Hodge-$\mathcal D$-Conjecture for the constructed regulator cycles

Let $d$ be the degree parameter of the family considered in the source, let $X_t$ be the corresponding fiber, and let $\mathscr{I}_0$, $\mathscr{I}_{\infty}$, and $\mathscr{D}$ be the specified collections of higher cycles. Let $r^{2,1}_{\mathcal{D},\mathbb{R}}$ denote the real regulator and let $H^{1,1}_{\mathbb{R}}(X_t)(1)$ be the real $(1,1)$-part with Tate twist. Hodge-$\mathcal D$-Conjecture. For general $d$, the image $r^{2,1}_{\mathcal{D},\mathbb{R}}(\mathscr{I}_0\cup\mathscr{I}_{\infty}\cup\mathscr{D})$ spans $H^{1,1}_{\mathbb{R}}(X_t)(1)$; hence the Hodge-$\mathcal D$-Conjecture holds in this case. The preceding discussion explains that the assertion is intended to extend the explicit regulator computation beyond the cases treated directly. The supplied source does not establish whether this general-$d$ assertion is proved or remains conjectural.