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.