The points-and-lines conjecture for free and K(pi,1) arrangements

Let ${\mathcal A}$ be a hyperplane arrangement with underlying matroid $G$. The points of $G$ are its flats of rank one, and its lines are its flats of rank two. Saying that $G$ is determined by its points and lines means that the matroid structure is uniquely determined by these rank-one and rank-two flats.

Points-and-lines conjecture. The underlying matroid of a free or $K(\pi,1)$ arrangement is determined by its points and lines.

This conjecture would imply that the combinatorial structure of matroids underlying free or $K(\pi,1)$ arrangements is controlled by rank-one and rank-two data. The text connects it to Terao’s freeness conjecture and the matroidality of the homotopy type of arrangement complements, but supplies no resolution.