Jacobi-type theorem for the vanishing intermediate Jacobian

Let $X$ be a smooth hyperplane section of a $2n$-dimensional complex projective manifold. Let $\mathbb{G}_X$ be the desingularization of the compactification of the connected covering space associated with the stabilizer of an elementary vanishing cycle, and let $J_{\operatorname{van}}(X)$ be the vanishing intermediate Jacobian. For $k\geq 1$, define

$$\mathbb{G}_X^{(k)}:=\mathbb{G}_X\times_{\mathbb{O}_X}\mathbb{G}_X\times_{\mathbb{O}_X}\cdots\times_{\mathbb{O}_X}\mathbb{G}_X.$$

Jacobi-type theorem. There is some positive integer $k$ such that the topological Abel--Jacobi mapping

$$\operatorname{a}_X^{\operatorname{top}}:\mathbb{G}_X^{(k)}\to J_{\operatorname{van}}(X)$$

is surjective for all $k\geq K$.

This conjecture predicts that sufficiently many copies of the parameter space for elementary vanishing cycles fill the vanishing intermediate Jacobian. It is stated as an open problem in the paper, and no resolution is supplied in the excerpt.