Full Hodge-isogeny property of the octic-to-dodecic K3 maps

Determine whether, for every magic octic K3 surface and every associated magic dodecic K3 surface, the induced rational map between them yields an isogeny of their full Hodge structures.

Background

The paper proves that the rational maps from magic octic K3 surfaces to associated magic dodecic K3 surfaces induce rational Hodge isometries on the transcendental parts, and consequently establish twisted derived equivalence after resolution.

The authors then explicitly ask whether the stronger statement holds for the entire Hodge structures, not merely their transcendental summands. This stronger isogeny assertion is left unresolved.

References

We wonder whether for any given choice of magic octic K3 surface $S$ and any choice of associated magic dodecic K3 surface $S'$, the rational map $S \dashrightarrow S'$ yields an isogeny of the full Hodge structure.

— The algebraic geometry of 3-by-3 magic squares of squares  (2609.09351 - Auel et al., 8 Sep 2026) in Section “Product Coordinate Projections: Magic Dodecic K3s,” immediately after Proposition 4.5