Koszul but non-G-quadratic Orlik–Solomon algebras

Construct, or determine the existence of, a central complex hyperplane arrangement whose Orlik–Solomon algebra is Koszul but not G-quadratic.

Background

G-quadraticity means that the Orlik–Solomon ideal admits a quadratic Gröbner basis after a linear change of basis in the presenting exterior algebra. The paper proves that parallel connections of supersolvable matroids yield G-quadratic Orlik–Solomon algebras, but the authors do not know whether their other constructions preserve this stronger property.

The problem therefore asks whether Koszulness can occur strictly beyond G-quadraticity within the class of Orlik–Solomon algebras.

References

We do not know if the same is true for our other mechanisms. The natural next question is then the following.

\begin{Qst} Is there a central arrangement $\A$ whose Orlik--Solomon algebra is Koszul but not G-quadratic? \end{Qst}

— Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements  (2609.03836 - Le et al., 3 Sep 2026) in Section 6, Questions; paragraph preceding the fourth Question