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tokens: modes are combined. Had that study examined the case of ω = 90°, it would have found smaller widths showing that the dynamics is more complex than just summing up single mode widths. This fact was later recognized in a more recent article (Gallardo 2020 ). Secondly, our analytical disturbing function as well as our multiple-argument pendulum models not only give precise resonance widths for each mode, their combinations, and their variations as functions of ω, they also agree with the chaos indicator. So our approach is not just to compare the analytical widths to the integration of the full equations of motion like it is done in the ‘dynamical map’. Instead, we seek the numerical separatrices obtained from measuring precisely how two nearby orbits diverge. In this sense, it is worth reminding the reader that the megno chaos indicator is one of the most precise diagnostic tools in resonance dynamics. In conclusion, the fact that the disturbing function is an expansion with respect to eccentricity (as in this work we used the second interpretation of the disturbing function without expanding with respect to inclination like in section 5.3 of Paper I) does not mean that it cannot probe the dynamics accurately. For instance, we could even explain for the first time the width of the chaotic separatrix layer of the inner 2:1 resonance as well as predict accurately the extent of the asymmetric libration domains for outer resonances. Therefore, the conclusion of Gallardo ( 2019 ) regarding the importance of the disturbing function at arbitrary inclinations and the analytical pendulum models is unjustified. While in the process of developing our multi-argument two-harmonic pendulum model to derive resonance widths for outer as well as inner resonances, we learned of the work of Lei ( 2019 ). The author tested the conclusion of Gallardo ( 2019 ) that the expansion of the disturbing function cannot model the 3:1 resonance accurately and developed what they termed a ‘multi-harmonic pendulum model’ and applied it to the disturbing function of Paper I. The author says that the disturbing function they use is somewhat different but it is not as they choose from the outset not to expand the perturbing potential with respect to inclination. That possibility is already present in Paper I and was termed the second interpretation of the disturbing function in section 4.3 of Paper I. It was already used to estimate resonance widths in section 5.3 of Paper I and was reprised in this paper. What Lei ( 2019 ) does differently is to rewrite the fully derived disturbing function (3) by trigonometrically splitting the resonant argument with respect to as follows: The details of the indices in the above sums are found in Lei’s paper and are not essential to our discussion. The author then selects from this new expression the ‘harmonics’ of in order to model the 3:1 resonance with the disturbing terms: We use the quotes to refer to harmonics in Lei’s formulation in