Beilinson's injectivity conjecture for Deligne cycle classes

Let $X$ be a smooth projective variety defined over a number field $k$, and let $p$ be a positive integer. Write $X_{\mathbb{C}}:=X\times_k\mathbb{C}$, and let $r$ be the Deligne cohomology cycle class map

$$CH^{p}(X_{\mathbb{C}})_{\mathbb{Q}}\xrightarrow{r}H_{\mathcal{D}}^{2p}(X_{\mathbb{C}},\mathbb{Z}(p))_{\mathbb{Q}}.$$

Beilinson's conjecture. The composition

$$CH^{p}(X)_{\mathbb{Q}}\longrightarrow CH^{p}(X_{\mathbb{C}})_{\mathbb{Q}}\xrightarrow{r}H_{\mathcal{D}}^{2p}(X_{\mathbb{C}},\mathbb{Z}(p))_{\mathbb{Q}}$$

is injective; equivalently, every class whose image in Deligne cohomology vanishes is zero. This is a difficult case of the Bloch–Beilinson philosophy; the source reports that no example of dimension at least $2$ with large Chow ring was known there, while the paper proves only its infinitesimal form.