Geometric height pairing and Lear extension conjecture

Let $X/S$ be a smooth projective family and let $\overline{X}/\overline{S}$ be a semistable degeneration over a partial compactification $\overline{S}\supset S$. Let $Z,W$ be algebraically trivial cycles whose Chow classes are $z,w$, and assume that $Z,W$ admit admissible liftings $Z^h,W^h$ on $\overline{X}/\overline{S}$. Let $[\mathcal{B}_{Z,W},||.||]_{\overline{S}}$ denote the Lear extension of the biextension line bundle and let $\mathcal{E}_{\langle z,w\rangle}$ be the geometric height-pairing line bundle.

Geometric height pairing and Lear extension conjecture. There is a canonical isomorphism of $\mathbb{Q}$-line bundles over $\overline{S}$:

$$[\mathcal{B}_{Z,W},||.||]_{\overline{S}}\cong \mathcal{E}_{\langle z,w\rangle}.$$

This conjecture identifies the archimedean biextension metric's Lear extension with the algebraic line bundle encoding the geometric Beilinson--Bloch height pairing, and would explain the degeneration of the archimedean height in terms of non-archimedean intersection theory. The paper proves the comparison for algebraically trivial cycles under Griffiths's conjecture on incidentally trivial cycles.