the inner and outer long-axis tubes using the criteria from @gaj15. For an orbit to be classified as an inner tube, the absolute value of the $x$-coordinate of the particle when it reaches the maximum value of $r_{yz}$ must be larger than the corresponding value when $r_{yz}$ has a minimum, that is, $\left|x(r_{yz,\mathrm{max}})\right| >\left|x(r_{yz,\mathrm{min}})\right| $. For the outer tubes, the inverse condition is fulfilled: $\left|x(r_{yz,\mathrm{max}})\right| <\left|x(r_{yz,\mathrm{min}})\right| $.

We find that indeed the outer tubes are more likely to be found in the upper branch of the $\kappa(\omega)$ plot while the inner tubes are typically situated in the lower one. By visual inspection of some orbits we see that this rule does not identify the tubes correctly in all cases as particle orbits are not always regular: they are still being disturbed by perturbations related to the past merger or by the time-dependent potential due to figure rotation. Fig. \[fig:porb\] shows two examples of orbits from the upper branch of $\kappa(\omega)$ in the top row and from the lower branch in the bottom row that were classified as the outer and inner long-axis tubes respectively. Clearly, the shape of the orbits is as expected for these two sub-families, i.e. the outer 