Infinite generation of invariant Griffiths groups for the cubic sevenfold and weighted quartic families

Let $X_3$ and $X_4$ be general members of the families of cubic sevenfolds and smooth hypersurfaces in $\mathbb{P}(1^6;2)$ described above, respectively. Write ${\rm Griff}_{\mathbb{Q}}(X_3)$ and ${\rm Griff}_{\mathbb{Q}}(X_4)$ for their Griffiths groups with rational coefficients, and let the displayed finite groups act as in the constructions above. Infinite-generation conjecture. For the general member of each family, the invariant Griffiths groups

$${\rm Griff}_{\mathbb{Q}}(X_3)^{\mathbb{Z}_3 \oplus \mathbb{Z}_3}\quad\text{and}\quad {\rm Griff}_{\mathbb{Q}}(X_4)^{\mathbb{Z}_2}$$

are infinitely generated. These examples relate Griffiths groups of products of a K3 surface or elliptic curve to invariant Griffiths groups of Fano hypersurfaces; the conjecture predicts that the resulting invariant groups are nevertheless infinitely generated for general members of the two families.