Height comparison conjecture for nodal odd-dimensional hypersurfaces

Let $X_0$ be an odd-dimensional geometrically integral projective hypersurface of degree at least three over a number field, with a single nodal singularity. Let $Q$ be the exceptional quadric in the blow-up of $X_0$ at the node, and let the two different rulings of $Q$ determine a difference whose Beilinson--Bloch height is defined. Let a smoothing deformation of $X_0$ determine a limit mixed Hodge structure with biextension height. Height comparison conjecture. The following two real numbers coincide, up to an element of $\mathbb{Q} \cdot \log |\mathbb{Q}^\times|$: the Beilinson--Bloch height of the difference of the two different rulings on $Q$ and the biextension height of the limit mixed Hodge structure determined by any smoothing deformation of $X_0$. This conjecture relates an arithmetic height associated with the exceptional quadric to a Hodge-theoretic height of the degeneration; the cited works provide the relevant notions, while the supplied source does not state a resolution.