The isotropic-subspace conjecture for hyperplane arrangements

Let $A$ be a hyperplane arrangement in $\mathbb{C}\mathbb{P}^n$, and let $V$ be an $n$-isotropic subspace of its Orlik–Solomon algebra of dimension $d\geq n$ that is not contained in any coordinate hyperplane. The isotropic-subspace conjecture. The arrangement is a union of $d+1$ fibres of a linear system of hypersurfaces of dimension $n-1$. The result is proposed for every arrangement; the paper's methods do not appear sufficient to prove it. It generalizes the known correspondence between isotropic subspaces and supported linear systems for certain line arrangements and arrangements with isolated non-normal crossings.