Intrinsic recovery of the six Schur quartics and line blocks

Determine whether the six Schur quartic surfaces, their partition into two triples, and the eleven 16-line blocks can be recovered uniquely from the abstract 176-line multi-incidence geometry without using the defining equations, thereby making the $S_3\times S_3$ action an intrinsic symmetry of that incidence geometry.

Background

The paper constructs six Schur quartics in two triples and identifies eleven 16-line blocks in their union. The induced permutation group on the six surfaces is S3×S3S_3\times S_3.

The unresolved canonicity question is whether these surfaces, their partition, and the block decomposition are determined solely by the abstract incidence geometry of the 176 lines rather than by the explicit equations used in the construction.

References

Can the six surfaces, their partition into two triples, and the eleven $16$-line blocks be recovered uniquely from the abstract $176$-line multi-incidence geometry, without using the defining equations~(4.1) and~(4.3)? A positive answer would make the $S_3\times S_3$ action an intrinsic symmetry of the incidence geometry itself.

— The Reye geometry inside the 64 lines of the Schur quartic  (2609.10751 - Nurowski, 9 Sep 2026) in Section 5, Conclusions and open problems, item 5