The quasi-isomorphism refinement of the Orlik–Solomon conjecture

Let $\mathcal{L}$ be a geometric lattice. Let $\mathrm{OS}(\mathcal{L})$ be its Orlik–Solomon algebra, and let $\mathbb{I}_{\mathcal{L}}$ be the map defined in the source. Quasi-isomorphism refinement. The following statements are equivalent:

1. $\mathcal{L}$ is supersolvable.
2. $\mathrm{OS}(\mathcal{L})$ is Koszul.
3. $\mathbb{I}_{\mathcal{L}}$ is a quasi-isomorphism.

This is proposed as a refinement of the classical Orlik–Solomon Koszulness conjecture. The source does not give a resolution, so the refinement remains open.