Cambrian fan polytopality conjecture

Let $W$ be a finite Coxeter group and let $\stackrel{\to}{G}$ be an orientation of its Coxeter diagram. The Cambrian fan

$${\mathcal F}(\stackrel{\to}{G})$$

is the fan whose maximal cones are the unions of reflecting chambers corresponding to the congruence classes of the Cambrian congruence. Cambrian fan polytopality conjecture. The fan ${\mathcal F}(\stackrel{\to}{G})$ is the normal fan of a simple convex polytope. If $W$ is crystallographic, this polytope is combinatorially isomorphic to the generalized associahedron for $W$. This conjecture predicts a polytopal realization of every Cambrian fan and identifies it with the generalized associahedron in the crystallographic case. The paper presents weaker related conjectures and proves the crystallographic associahedral statement in types $A$ and $B$.