Forbidden-subgraph characterization of distributive Cambrian lattices

Let $W$ be a finite Coxeter group, let $\gamma\in W$ be a Coxeter element, and let $\Gamma_{\gamma}(W)$ be the orientation of the Coxeter diagram of $W$ induced by $\gamma$. Assume that $\Gamma_{\gamma}(W)$ does not contain any of the induced subgraphs listed in Proposition 1.6 of the source. Let $\mathcal{B}_{\gamma}$ denote the corresponding $\gamma$-Cambrian lattice.

Forbidden-subgraph conjecture. The list in Proposition 1.6 is exhaustive: if $\Gamma_{\gamma}(W)$ contains none of the listed induced subgraphs, then $\mathcal{B}_{\gamma}$ is distributive.

This conjecture proposes a complete combinatorial criterion for distributivity of the relevant Cambrian lattices in finite Coxeter groups. The notation for $\mathcal{B}_{\gamma}$ and the exact forbidden-subgraph list are given in the cited proposition.