Classification conjecture for free multi-braid arrangements

Let $(A_\ell,\mathbf{m})$ be a multi-braid arrangement, and let a free ANN multiplicity mean a multiplicity of the type specified in the preceding results. A free vertex is a vertex satisfying the freeness condition used in Theorem~. The balanced cone of multiplicities is the cone referred to in Theorem~.

Classification conjecture. The multi-braid arrangement $(A_\ell,\mathbf{m})$ is free if and only if it is one of the multi-braid arrangements constructed in Corollary~. Equivalently, if $(A_\ell,\mathbf{m})$ is free, then either $\mathbf{m}$ is a free ANN multiplicity or $\mathbf{m}$ has a free vertex. Equivalently, if $\mathbf{m}$ is a free multiplicity which is not in the balanced cone of multiplicities, then $\mathbf{m}$ has a free vertex.

This conjecture would classify all free multi-braid arrangements in terms of the two inductive constructions given by Corollary~. Its status is not resolved in the supplied text.