Beilinson's Hodge-index conjecture for height pairings

Let $X$ be a smooth projective variety of dimension $d$ over a number field $k$, and let $L_X\in\operatorname{CH}^1(X)$ be the class of a hyperplane section. Assume the hard Lefschetz conjecture on null-homologous cycles and consider the primitive cycle decomposition. Let $\langle\,\ \rangle_{\rm HT}$ be the height pairing. Beilinson's Hodge-index conjecture. For $r\leq\frac{d+1}{2}$, the form

$$\langle\,L^{d-2r+1}\ \rangle_{\rm HT}$$

is definite of sign $(-1)^r$ on the primitive $r$-cycles. This is the height-pairing analogue of the Hodge index theorem and depends on the stated hard Lefschetz conjecture; it remains open in the supplied source.