Beilinson's non-degeneracy conjecture for the height pairing

Let $X$ be a smooth projective variety of dimension $d$ over a number field $k$. Write $\operatorname{CH}^r_{\hom}(X;\mathbb{Q})$ for homologically trivial codimension-$r$ cycles modulo rational equivalence. Under the assumptions under which Beilinson's height pairing is defined, let

$$\langle\,\ \rangle_{\rm HT}:\operatorname{CH}^r_{\hom}(X;\mathbb{Q})\times\operatorname{CH}^{d-r+1}_{\hom}(X;\mathbb{Q})\longrightarrow\mathbb{R}$$

be that pairing. Beilinson's non-degeneracy conjecture. The height pairing is non-degenerate. This is one of Beilinson's conjectural properties of the arithmetic height pairing and remains open in the supplied source.