 Solution

True. Slide beads upward separately in each of the refined residue classes modulo $ef$. Grouping the refined runners first by residue modulo $f$ and then packing each component modulo $e$ produces $C_e(Q_f(\lambda ))$. Packing modulo $ef$ before forgetting the refinement produces $Q_f(C_{ef}(\lambda ))$. These are the same runner configurations, so