The integral-roots characterization of free connected subgraph arrangements

Let $G$ be a connected graph, let $\mathcal{A}_G$ be its connected subgraph arrangement, and let $\chi(\mathcal{A}_G)$ denote the characteristic polynomial of this arrangement. Integral-roots conjecture. The polynomial $\chi(\mathcal{A}_G)$ has only integral roots if and only if $\mathcal{A}_G$ is free. The paper states that, beyond its classification of free connected subgraph arrangements, no characterization of graphs with this integral-root property is known; the equivalence therefore remains open.