  This paper shows the polytopality of any finite type $\mathbf{g}$-vector fan,
acyclic or not. In fact, for any finite Dynkin type $\Gamma$, we construct a
universal associahedron $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$ with the property
that any $\mathbf{g}$-vector fan of type $\Gamma$ is the normal fan of a
suitable projection of $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$.
