Oriented matroids from type $\mathbb{A}$ cluster categories

For any cluster-tilting object $\mathsf{T}$ in the cluster category ${\mathcal{C}}_n$ of type ${\mathbb{A}}_n$, we construct a rank-four oriented matroid ${\mathcal{M}}_{\mathsf{T}}$ such that stackable triangulations of ${\mathcal{M}}_{\mathsf{T}}$ are in bijection with equivalence classes of maximal green sequences with initial cluster $\mathsf{T}$. This generalises the result that equivalence classes of maximal green sequences of linearly oriented ${\mathbb{A}}_n$ are in bijection with triangulations of a three-dimensional cyclic polytope. The definition of the oriented matroid ${\mathcal{M}}_{\mathsf{T}}$ arises from the extriangulated structure on ${\mathcal{C}}_n$ which makes $\mathsf{T}$ projective.