Koszulness of OS(L) iff supersolvability

Prove that for every geometric lattice L, the Orlik–Solomon algebra OS(L) is Koszul if and only if L is supersolvable.

Background

Yuzvinsky conjectured that supersolvability of L characterizes when OS(L) is Koszul. The present paper proves Koszulness for supersolvable L via the MD(L) model, but the converse direction remains open in general.

This conjecture sits at the intersection of matroid theory, arrangement theory, and Koszul algebra theory, and is a central long-standing problem.

References

To finish let us mention the following classical conjecture. The Orlik--Solomon algebra $OS(L)$ of a geometric lattice $L$ is Koszul if and only if $L$ is supersolvable.

— Matroid complexes and Orlik-Solomon algebras  (2506.15048 - Coron, 18 Jun 2025) in Section 6 (Koszulness of Orlik--Solomon algebras), concluding subsection

If $\OS(\A)$ is Koszul, must $\pi(\A,t)$ factor into linear factors?

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

We were however unable to deduce this using the mechanisms described in this paper. This leads us to ask the following.

\begin{Qst} Is the Betsy Ross matroid OS-Koszul?\footnote{That the Betsy Ross matroid might be OS-Koszul was first pointed out to the authors by Peeva.} \end{Qst}

— Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements  (2609.03836 - Le et al., 3 Sep 2026) in Section 6, Questions; paragraph containing the Betsy Ross matroid question