Covering graph conjecture for token graphs of odd stars

Let $K_{1,n}$ be the star graph with $n$ leaves, let $n$ be odd, and set $k=(n+1)/2$. A combined voltage assignment $(\alpha,\omega)$ on a base graph $X$ with values in the group $\mathbb{Z}_{2n}$ defines a covering graph $X^{(\alpha,\omega)}$.

Covering conjecture for odd-star token graphs. The token graph $F_k(K_{1,n})$ is isomorphic to a covering graph $X^{(\alpha,\omega)}$ of a base graph $X$ on

$$\frac{1}{2n}\binom{2k}{k}$$

vertices, with combined assignment on the group $\mathbb{Z}_{2n}$.

The preceding discussion establishes the construction in particular cases, including $n=3$ and $n=5$, while the conjecture proposes it for every odd $n$.