Top panel: stars part of the substructures identified by Dodd et al. (2023) in (E, L_z, L_⊥) space at the present day, colour-coded by t_λ. Stars with t_λ > 13.7 Gyr are shown in black. The light grey stars in the background correspond to the whole local halo sample. A number of substructures and regions of (E, L_z, L_⊥) space are labelled in the leftmost panel. Middle panel: same as the top panel, but after 10 Gyr of integration time in the fiducial potential, and only showing stars whose final three apocentres were on average larger than 7 kpc. Bottom panel: also after 10 Gyr of integration time, but showing (E_J, L_z) and (E_J, L_⊥) instead of E. We have indicated the line E_J = E−Ω_b L_z for E = −25 000 km² s⁻² (corresponding to the upper limit in energy of the top and middle panel) and E = 0 km² s⁻² (corresponding to unbound stars in our potential).