The generic local-system cohomology conjecture for Orlik–Solomon algebras

Let $M$ be the complement associated with the arrangement, let $A^*$ be its Orlik–Solomon algebra, let $A_1^*$ be the parameter space of weights, and let ${\cal L}(a)$ be the local system on $M$ associated with $a\in A_1^*$. For each integer vector $N\in{\mathbb Z}^n$, write $a+N$ for the corresponding shifted weight.

Generic local-system cohomology conjecture. For every $p$ and almost all $a\in A_1^*$ among those with

$$H^p(M,{\cal L}(a))\ne 0$$

one has

$$\dim H^p(M,{\cal L}(a))=\sup_{N\in{\mathbb Z}^n}\dim H^p(A^*,a+N).$$

The preceding inequality shows that the left-hand side is at least each term in the supremum, and the conjecture asserts equality for almost all parameters with nonvanishing local-system cohomology. The supplied text gives no resolution, so the conjecture remains open.