The equivalence conjecture for arrangement complements and boundary manifolds

Let ${\mathcal{A}}$ and ${\mathcal{A}}'$ be central arrangements in $\mathbb{C}^3$. Write $U({\mathcal{A}})$ for the complement, ${\partial{{\overline{U}}}}({\mathcal{A}})$ for its boundary manifold, $\Delta_{{\partial{{\overline{U}}}}({\mathcal{A}})}(t)$ for the corresponding Alexander polynomial, $\Gamma({\mathcal{A}})$ for the graph associated to the arrangement, and $L({\mathcal{A}})$ for its intersection lattice. Arrangement-complement equivalence conjecture. The following conditions are equivalent: $U({\mathcal{A}})\cong U({\mathcal{A}}')$; ${\partial{{\overline{U}}}}({\mathcal{A}})\simeq {\partial{{\overline{U}}}}({\mathcal{A}}')$; $\Delta_{{\partial{{\overline{U}}}}({\mathcal{A}})}(t)=\Delta_{{\partial{{\overline{U}}}}({\mathcal{A}}')}(t)$; $\Gamma({\mathcal{A}})\cong\Gamma({\mathcal{A}}')$; and $L({\mathcal{A}})\cong L({\mathcal{A}}')$.

This is proposed as a refinement of the Jiang–Yau conjecture. The examples in the source show that non-isomorphic intersection lattices can yield non-homotopy-equivalent boundary manifolds, while the full equivalence of the five conditions remains open.