Carter–Marsh conjecture on simplicial regions of Lusztig's piecewise-linear function

Suppose $\mathbf{g}$ is of type $A_n$, and let $\mathbf{i}$ be a reduced expression for the longest Weyl-group element $w_0$. Let $\mathbf{k}=(1,3,5,\ldots,2,4,6,\ldots)$ and $\mathbf{k'}=(2,4,6,\ldots,1,3,5,\ldots)$ be the opposite reduced expressions for $w_0$. For $j=1,2,\ldots,n$ and for $P$ a partial quiver of type $A_n$ associated with $\mathbf{i}$, let $c_j(\mathbf{k})$ and $c_P(\mathbf{k})$ be the corresponding Lusztig parametrizations, and let $R_{\mathbf{k}}^{\mathbf{k'}}$ be Lusztig's piecewise-linear transition function. Carter–Marsh conjecture. The vectors $c_j(\mathbf{k})$, $j=1,2,\ldots,n$, and $c_P(\mathbf{k})$, with $P$ a partial quiver associated with $\mathbf{i}$, form a simplicial region of linearity $X_{\mathbf{i}}$ of $R_{\mathbf{k}}^{\mathbf{k'}}$. Furthermore, all such simplicial regions arise in this way, and the map $\mathbf{i}\mapsto X_{\mathbf{i}}$ from the graph of commutation classes of reduced expressions for $w_0$, with edges given by the long braid relation, to the graph of simplicial regions of linearity of $R$, with edges given by adjacency, is a graph isomorphism. The conjecture describes the full simplicial decomposition of the piecewise-linear transition map in type $A_n$, linking commutation classes of reduced expressions with its regions of linearity; the supplied text says it will be shown in type $A_4$, while the general assertion remains unresolved in this source.