Intersection-lattice conjecture for the defect of line arrangements

For a reduced plane curve $C=\{f=0\}$ in $\mathbb{P}^{2}_{\mathbb{C}}$, let $\nu(C)$ denote its defect. A line arrangement is a union of finitely many lines in $\mathbb{P}^{2}_{\mathbb{C}}$, and its intersection lattice records the intersections among its lines and the corresponding incidence relations. Intersection-lattice conjecture. For a line arrangement $\mathcal{L} \subset \mathbb{P}^{2}_{\mathbb{C}}$, the defect $\nu(\mathcal{L})$ is determined by the intersection lattice of $\mathcal{L}$; more precisely, if $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are two line arrangements with isomorphic intersection lattices, then

$$\nu(\mathcal{L}_{1})=\nu(\mathcal{L}_{2}).$$

This is presented as a difficult open problem and as a broad generalization of Terao's freeness conjecture for line arrangements.