Simpliciality conjecture for bipartite biCambrian fans

Let $W$ be a finite Coxeter group and let $c$ be a bipartite Coxeter element of $W$. The fan $\operatorname{biCamb}(W,c)$ is the bipartite biCambrian fan.

Simpliciality conjecture. The fan $\operatorname{biCamb}(W,c)$ is simplicial. Equivalently, its dual polytope is simple.

The paper proves this conjecture in types A and B. It is motivated by the fact that the corresponding fan is the normal fan of a polytope, while simpliciality fails for linear Coxeter elements in type A.