Connectedness conjecture for friends-and-strangers graphs of complete bipartite graphs

Let $Y$ be a graph on $n\ge 2k\ge 4$ vertices. A $k$-bridge is a set of $k$ edges whose deletion disconnects the graph, and it is non-trivial when neither resulting component is an isolated vertex. The graph $\mathsf{FS}(K_{k,n-k},Y)$ is the friends-and-strangers graph associated with the graphs $K_{k,n-k}$ and $Y$. Connectedness conjecture. The graph $\mathsf{FS}(K_{k,n-k},Y)$ is connected if and only if $Y$ is a connected non-bipartite graph without a non-trivial $k$-bridge and $Y\not=C_n$. This conjecture extends the established disconnectedness criterion, while the cited theorems provide only weaker sufficient conditions for connectedness; the full characterization remains open.