Height inequality conjecture for Beilinson–Bloch heights

Let $S$ be a quasi-projective variety over a number field $K$, let $f\colon X\to S$ be a smooth projective family with relatively ample line bundle $N$, and let $Z$ be the family of cycles defining the normal function and Betti stratum $S^{\mathrm{Betti}}(1)$. Set

$$S^{\mathrm{amp}}=S\setminus S^{\mathrm{Betti}}(1).$$

A function on $S(\overline{\mathbb{Q}})$ is a dominant height on an open subvariety $U$ if it satisfies the stated lower bound against a logarithmic Weil height on $U$. Height inequality conjecture. The Beilinson–Bloch height function $s\mapsto\langle Z_s,Z_s\rangle_{\mathrm{BB}}$ is a dominant height on $S^{\mathrm{amp}}$. Moreover, $S^{\mathrm{amp}}$ is the largest open subvariety of $S$ on which this function is a dominant height. This predicts both a height lower bound away from the Betti stratum and the maximality of its domain; the source presents it as a conjecture and gives no resolution.