The line-closed matroid–quadratic Orlik–Solomon conjecture

Let $G$ be a matroid, and let $A(G)$ be its Orlik–Solomon algebra. The matroid $G$ is line-closed if every subset closed under the closure of each pair of its elements is a flat, and $A(G)$ is quadratic if it has a presentation whose relations all have degree two.

Line-closed matroid–quadratic Orlik–Solomon conjecture. $G$ is line-closed if and only if $A(G)$ is quadratic.

The conjecture proposes a purely matroidal characterization of quadratic Orlik–Solomon algebras. The surrounding discussion identifies line-closure as a combinatorial analogue of formality, but the supplied text gives no resolution of the equivalence.