Classification of conics on the magic-square surface

Determine whether the 128 rational conics arising from the three-distinct-entry locus are all the conics contained in the surface V parameterizing 3 × 3 magic squares of squares.

Background

The appendix proves that V contains no lines, while the body of the paper constructs many conics, including 128 rational conics arising from the locus of magic squares with three distinct entries. These conics lie entirely in the nondistinct locus and therefore do not directly resolve the Diophantine problem for distinct entries.

The authors note that an external result implies that only finitely many conics occur, but they do not classify all of them. They explicitly leave open whether the known 128 conics exhaust the conics on V.

References

We leave open the question of whether these are all the conics contained in $V$; we know there are finitely many by .

— The algebraic geometry of 3-by-3 magic squares of squares  (2609.09351 - Auel et al., 8 Sep 2026) in Appendix, Section “Lines on V,” final remark