The disjoint-path connectivity conjecture for friends-and-strangers graphs with a star

Let $X$ be a graph on $n$ vertices, and let $\Star_n$ denote the star graph on $n$ vertices. Write

$$G=\operatorname{FS}(X,\Star_n),$$

and assume that $G$ is connected. For permutations $\sigma$ and $\rho$ in $V(G)$, the maximum number of disjoint paths between $\sigma$ and $\rho$ is

$$\min\{\deg(\sigma),\deg(\rho)\}.$$

Disjoint-path connectivity conjecture. The maximum number of disjoint paths between any two permutations $\sigma$ and $\rho$ in $G$ equals $\min\{\deg(\sigma),\deg(\rho)\}$. This would determine the local vertex connectivity between every pair of vertices in this class of friends-and-strangers graphs, extending the paper's connectivity results and the evidence provided by starcle graphs. The conjecture is presented as a future research direction, and its resolution is not given here.