Before revision:  We  complement these positive results by showing that, for any constant p \geq 4, both problems are W[1]-hard on C_\ell\dots -free graphs, where 4 \leq \ell \leq p, and Token Sliding remains W[1]-hard even on bipartite graphs.
After revision:  We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant p \geq 4, both problems are W[1]-hard on C_\ell\dots -free graphs, where 4 \leq \ell \leq p, and Token Sliding remains W[1]-hard even on bipartite graphs.