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tokens: require high sodium abundances at low oxygen abundances. In the following, we list the three quantitative discrepancies between AGB nucleosynthesis yields and observed abundance patterns that are solved if we allow for smaller mass-loss rates. (1) The oxygen depletion in the most massive super-AGBs is not large enough. To address this problem, we have invoked deep mixing in the giant progenitors of the most extreme population, a mixing which should be favoured by the large helium abundance of these stars (D'Antona Ventura 2007 ), and we adopt this solution also here (see Section 4.1 ). Nevertheless, smaller mass-loss rates in the super-AGB phase would lead to the required large depletion of oxygen and would provide an additional possible solution to this problem. (2) Apart from 24 Mg depletion, some other HBB proton captures require a longer time than allowed by the mass-loss rates chosen in our Ventura et al. ( 2013 ) paper and in the previous computation. In particular, the silicon increase by proton captures on aluminium is found in Ventura et al. ( 2013 ) at the level of ∼0.05 dex, while ∼0.15 dex are required by Carretta ( 2015 ) data for NGC 2808. Also in this case, a larger production requires longer evolutionary times. (3) Taking at face value the modest δ Y implied by the observational width of the MS of some clusters in recent observations, Bastian, Cabrera-Ziri Salaris ( 2015 ) found a discrepancy between the range covered by data in O–Na plane and the range allowed by models. Below, we discuss some limitation of this analysis, and other possible ways to overcome this problem. However, taking this result at face value, also in this case the difficulty is removed if the models have a larger oxygen depletion for a given T HBB , and this can be achieved assuming smaller mass-loss rates during the AGB evolution. If models with smaller T HBB can describe well the O–Na data, they are progeny of lower initial masses and have smaller Y yield, as required by the comparisons in Bastian et al. ( 2015 ), solving the discrepancy. The mass-loss rates in the AGB models by Ventura et al. ( 2013 ) were originally calibrated on a comparison with the luminous lithium-rich stars in the Magellanic Clouds (Ventura, D'Antona Mazzitelli 2000 ), but this calibration is also able to provide a positive sodium yield, in spite of the contemporary burning of sodium and oxygen. If we allow for a mass-loss rate smaller, say by a factor of 3 – that is, we triple the time for p-captures in the AGB envelope – we may find a solution for the four problems listed above, at the expenses of burning too much sodium (see e.g. Ventura et al. 2011 ; Doherty et al. 2014 ). The problem of the sodium destruction is examined in detail in Renzini et al. ( 2015 ), where it is suggested that a reduction by a factor of 5 of the reaction rate 23 Na(p,α) 20 Ne would allow sodium to be destroyed at a slower rate, so that its global yield may be in the range of the observed sodium