The converse supersolvability conjecture for ψ-graphical arrangements

Let $(G,\psi)$ be a graph together with the data defining its $\psi$-graphical arrangement $\mathcal{A}_{G,\psi}$. Suppose the vertices of $G$ can be ordered as $v_1,\dots,v_p$ so that each $v_{i+1}$ connects to previous vertices along a clique, and whenever $i<j$ and $v_i$ is adjacent to $v_j$, one has $\psi(v_j)\subseteq\psi(v_i)$. The arrangement is supersolvable when the intersection lattice of its cone contains a maximal chain of modular elements. The converse supersolvability conjecture. If $\mathcal{A}_{G,\psi}$ is supersolvable, then $(G,\psi)$ satisfies these two ordering conditions. The conjecture would extend the characterization of supersolvable graphical arrangements by chordal graphs to $\psi$-graphical arrangements; the paper notes that it may not be difficult to prove and suggests studying analogues of other characterizations of chordal graphs.