The classification conjecture for perfect colorings of infinite circulant graphs with odd distances

Let $k$ and $n$ be positive integers. Write $D_n$ for the set of the first $n$ positive odd integers, and let $\mathrm{Ci}_\infty(D_n)$ denote the infinite circulant graph whose vertices are the integers, with two vertices adjacent when their difference belongs to $D_n$ up to sign. A perfect $k$-coloring is a coloring with $k$ colors for which the number of neighbors of each color is determined solely by the color of the vertex.

Classification conjecture. The set of perfect $k$-colorings of $\mathrm{Ci}_\infty(D_n)$ consists of perfect colorings induced from perfect colorings of the infinite path graph and of the finite circulant graphs $\mathrm{Ci}_t(D_n)$ for $t=4n-2,4n,4n+2$.

The conjecture proposes that all perfect colorings in this family arise from the previously classified perfect colorings of the infinite path or from these three finite circulant graphs. The surrounding results establish periodicity of perfect colorings, but the stated classification is not resolved here.