Koszulness of the remaining quadratically generated examples

Determine which of the 150 remaining quadratically generated toric rings associated with the exceptional (0,1)-polytopes are Koszul.

Background

Among the 160 exceptional toric rings whose toric ideals are generated by quadrics, computational resolutions identify ten that are not Koszul. For the remaining 150 rings, the resolution is linear through homological degree four, but this computation does not settle Koszulness.

References

Their Koszulness remains open.

— Most $(0,1)$-polytopes are not normal  (2609.02778 - Morales, 2 Sep 2026) in Section 2, immediately following Proposition 2.8

Determine which of the remaining $150$ quadratically generated toric rings are Koszul. Does this family contain a Koszul toric ring with no quadratic Gröbner basis? Find a criterion in terms of the polytope, a flag unimodular triangulation, or the quadratic fibers.

— Most $(0,1)$-polytopes are not normal  (2609.02778 - Morales, 2 Sep 2026) in Problem Koszulness in a family of (0,1)-polytopes, Section 5, Open problems