Nef adelic extension conjecture for the Beilinson–Bloch biextension

Let $S$ be a variety over a number field $K$, and let $L$ be the biextension line bundle on $S$ whose adelic fiberwise structures have degrees equal to the Beilinson–Bloch heights where those heights are defined. Nef adelic extension conjecture. The line bundle $L$ has a natural extension to a nef adelic line bundle $\overline{L}$ on $S$, and for every $s\in S(\overline{\mathbb{Q}})$ where the Beilinson–Bloch height is well-defined, its restriction is the adelic line bundle $\overline{L}_s$. The conjecture would globalize the fiberwise adelic structures and their positivity; the source does not state that this extension has been constructed in general.