The forbidden-octahedral-graph characterization of gentle graphic arrangements

A graphic arrangement is the hyperplane arrangement $\mathcal{A}(G)$ associated with a graph $G$. Let $G_{\mathrm{oct}}$ denote the octahedral graph.

Gentle graphic-arrangement conjecture. A graphic arrangement $\mathcal{A}(G)$ is gentle if and only if $G_{\mathrm{oct}}$ cannot be obtained from $G$ by a series of edge contractions of an induced subgraph of $G$.

This conjecture proposes a forbidden-graph characterization of gentleness for graphic arrangements. The preceding results establish that gentleness is preserved under suitable localization and subarrangement operations, but the claimed equivalence is not resolved in the provided source.