Two-component conjecture for friends-and-strangers graphs with a complete bipartite factor

Let $Y$ be a connected bipartite graph on $n\ge 5$ vertices with no non-trivial cut edge and $Y\not=C_n$. The graph $\mathsf{FS}(K_{2,n-2},Y)$ is the friends-and-strangers graph associated with $K_{2,n-2}$ and $Y$. Two-component conjecture. The graph $\mathsf{FS}(K_{2,n-2},Y)$ has exactly $2$ components. Wilson's result establishes the analogous statement for $\mathsf{FS}(S_n,Y)$, and the cited theorem gives supporting evidence in the complete-bipartite case; the conjecture remains open.