Type A cluster-variable conjecture for strongly homogeneous modules

Let $\mathfrak{g}$ be of type $A_n$, let $I$ be the indexing set for the simple roots, and let $\mathcal{M}in_{0}^{+}$ denote the strict dominant minuscule elements of the Weyl group. For each total ordering $<$ on $I$, let $\mathcal{S}^{{\bf i}_{<}}$ be the corresponding standard seed, and let $S(w)$ be the strongly homogeneous module associated with $w$. Type $A$ cluster-variable conjecture. The set of cluster variables of the seeds $\mathcal{S}^{{\bf i}_{<}}$, with $<$ running over all possible orderings on $I$, is exactly the set of isomorphism classes of the strongly homogeneous modules $S(w)$ for $w\in\mathcal{M}in_{0}^{+}$. This extends the observed coincidence in types $A_3$ and $A_4$ between flag-minor cluster variables and prime strongly homogeneous modules; the general type $A_n$ assertion is left open.