Cambrian fan and cluster poset correspondence conjecture

Let $W$ be a finite crystallographic Coxeter group with root system $\Phi$, simple roots $\alpha_i$, fundamental weights $\omega_i$, and a bipartition $G=I_+\cup I_-$ of its Coxeter diagram. Set $\varepsilon(i)=+1$ for $i\in I_+$ and $\varepsilon(i)=-1$ for $i\in I_-$. The cluster fan is the fan whose maximal cones are generated by the clusters of roots, and the orientation $I_+\longrightarrow I_-$ directs every edge from $I_+$ to $I_-$. Cambrian cluster correspondence conjecture. The linear map $\alpha_i\mapsto\varepsilon(i)\omega_i$ maps the cluster fan for $W$ isomorphically to the Cambrian fan for the orientation $I_+\longrightarrow I_-$, and the Cambrian lattice for this orientation is the cluster poset. This identifies the geometric and order-theoretic Cambrian objects with the cluster fan and cluster poset. The statement is solved: the paper proves it in the relevant development, including the type-$A$ and type-$B$ cases discussed there.