Athanasiadis's freeness conjecture for coned deformations of braid arrangements

Let $\mathbb{K}$ be a field of characteristic zero, let $G$ be a directed graph on $V_G=\{1,2,\ldots,\ell+1\}$, and let $\mathcal{A}_G$ be the affine arrangement in $\mathbb{K}^{\ell+1}$ defined by

$$x_i-x_j=-\epsilon(i,j),0,1$$

for $1\leq i<j\leq \ell+1$, where $\epsilon(i,j)=1$ if $(i,j)\in E_G$ and $\epsilon(i,j)=0$ otherwise. Let $c\mathcal{A}_G$ denote the coning of $\mathcal{A}_G$. The graph $G$ satisfies (A1) if, for every triple $i,j,k$ with $i,j<k$, $(i,j)\in E_G$ implies $(i,k)\in E_G$ or $(k,j)\in E_G$, and it satisfies (A2) if, for every such triple, $(i,k)\in E_G$ and $(k,j)\in E_G$ imply $(i,j)\in E_G$; these conditions may hold after re-ordering $V_G$.

Athanasiadis's freeness conjecture. The coning $c\mathcal{A}_G$ is free if and only if $G$ satisfies (A1) and (A2).

This conjecture characterizes freeness for a family of deformations of the braid arrangement in terms of two directed-graph conditions. The stated status evidence says that conditions (A1) and (A2) were subsequently proved sufficient for the conjecture in a more general setting, so the conjecture is recorded as solved.