(Abridged) In the recent controversy about the role of TP-AGB stars in evolutionary population synthesis (EPS) models of galaxies, one particular aspect is puzzling: TP-AGB models aimed at reproducing the lifetimes and integrated fluxes of the TP-AGB phase in Magellanic Cloud (MC) clusters, when incorporated into EPS models, are found to overestimate the TP-AGB contribution in resolved star counts and integrated spectra of galaxies. In this paper, we call attention to a particular evolutionary aspect that in all probability is the main cause of this conundrum. As soon as stellar populations intercept the ages at which RGB stars first appear, a sudden change in the lifetime of the core He-burning phase causes a temporary boost in the production rate of subsequent evolutionary phases, including the TP-AGB. For a timespan of about 0.1 Gyr, triple TP-AGB branches develop at slightly different initial masses, causing their frequency and contribution to the integrated luminosity of the stellar population to increase by a factor of 2. The boost occurs just in the proximity of the expected peak in the TP-AGB lifetimes, and for ages of 1.6 Gyr. Coincidently, this relatively narrow age interval happens to contain the few very massive MC clusters that host most of the TP-AGB stars used to constrain stellar evolution and EPS models. This concomitance makes the AGB-boosting particularly insidious in the context of present EPS models. The effect brings about three main consequences. (1) Present estimates of the TP-AGB contribution to the integrated light of galaxies derived from MC clusters, are biased towards too large values. (2) The relative TP-AGB contribution of single-burst populations falling in this critical age range cannot be accurately derived by the fuel consumption theorem. (3) A careful revision of AGB star populations in intermediate-age MC clusters is urgently demanded.
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