Algebraic correspondence conjecture for the universal Calabi–Yau threefold construction

Let $F$ be an algebraically closed field, let $\E\to\mathbf{P}^1_F$ be a rational elliptic fibration with multiplicative fibers at $0$ and $1$ and an additive fiber at $\infty$, and let $\eta_3$ be the generic point of $(\mathbf{P}^1_F)^3$. Let $\caZ^3(\E)$ be the projective variety over $\eta_3$ defined by the fiber product in the source, and let $\E^3_{\eta_3}$ denote the corresponding triple fiber product. Algebraic correspondence conjecture. There exists an algebraic correspondence between $\caZ^3(\E)$ and $\E^3_{\eta_3}$ such that the induced map

$$H^3(\caZ^3(\E))\to H^3(\E^3_{\eta_3})$$

is surjective over the transcendental part of $H^3(\E^3_{\eta_3})$, with $H^3$ interpreted either as $\ell$-adic cohomology as a Galois representation or, when $F=\mathbf{C}$, as Betti cohomology in a family of Hodge structures. The conjecture is intended to reduce 3-modularity of these elliptic fibrations to an algebraic-cycle statement; the supplied text gives no resolution status.