(L_z, L_⊥) distribution after 8 Gyr of integration time of 319 particles with Helmi-Stream-like phase-space positions, selected from the recentred Progenitor 4 by Koppelman et al. (2019b), in a potential with a triaxial NFW halo for a few combinations of p and q. Figure E.1 shows this plot for an extended grid of p and q values. The colourbar indicates the variation in L_z over an integration time of 100 Gyr. In the axisymmetric potential, p = 1.00, L_z is an IoM and all particles are on z-tube orbits. For p > 1, the particles with larger L_⊥ are on y-tube orbits, meaning their L_z is no longer conserved, making them move towards lower L_z in time with respect to the particles on z-tube orbits. The timescale over which this happens depends on the degree of triaxiality of the potential, and in particular the value of p. A number of particles are strongly trapped by the Ω_ϕ : Ω_z = 1:1 resonance (selected as ΔL_{y, z} ≲ 750), and are encircled in green. A video, showing the behaviour of the particles in (L_z, L_⊥) space for a range of p and q over time, is available online.