Stasheff polytopes and the coordinate ring of the cluster $\mathcal{X}$-variety of type $A_n$

We define Stasheff polytopes in the space of tropical points of cluster $\mathcal{A}$-varieties.
We study the supports of products of elements of canonical bases for cluster $\mathcal{X}$-varieties.
We prove that, for the cluster $\mathcal{X}$-variety of type $A_n$, such supports are Stasheff polytopes.
No preliminary is required