Abel–Jacobi family formulation of the general Hodge conjecture

Let $X$ be a smooth projective variety of dimension $g$, and let $V\subset H^m(X,\mathbb{Q})$ be a $\mathbb{Q}$-Hodge substructure of level at most $m-2p$. A nonsingular projective family of subvarieties is a diagram

$$\begin{array}{ccc} \mathcal Z &\longrightarrow& X\\ \downarrow&&\\ S&& \end{array}$$

whose members have pure dimension $g-m+p$. Abel–Jacobi family formulation of the general Hodge conjecture. There exists such a family with base $S$ a nonsingular projective variety of dimension $m-2p$ such that the image of

$$H^{m-2p}(S,\mathbb{Q})$$

under the family's Abel–Jacobi map $q_*r^*$ contains $V$. The source says this is equivalent to the general Hodge conjecture; its general validity remains open.