The edge-coloring conjecture for dual graphs of alcoved triangulations

Let $G=G_{\mathcal{A},i,d}$ and $H=G_{r,d}$. The edges of $G$ are colored according to the types of hyperplanes from the affine Coxeter arrangement of type $A_{d+1}$ to which they correspond. Let $G\langle H\rangle$ denote the graph obtained by using connecting sets between copies of $H$ indexed by the vertices of $G$.

Edge-coloring conjecture. The edge-coloring of $G$ determined by the hyperplane types prescribes a choice of connecting sets such that

$$G\langle H\rangle$$

is isomorphic to the dual graph of the alcoved triangulation of $r\Delta_{i,d}$.

The claim proposes that hyperplane-type edge-coloring supplies the compatibility needed in the graph construction. The source presents it as a proposed candidate, and no resolution is supplied here.