Cluster factorization conjecture for outerplanar Landau diagrams

Let $\mathcal{L}=G_u\cup H_u^{\triangle}$ be a planar Landau diagram, where $G$ is an outerplanar graph, and let $\Delta_{\mathcal{L}}$ denote its LS discriminant. Cluster Factorization Conjecture. The LS discriminant $\Delta_{\mathcal{L}}$ factors into a product of cluster variables. The theorem preceding this conjecture establishes the analogous factorization when $G$ is a tree; the conjecture proposes the extension to all outerplanar graphs and is unresolved in the supplied text.