Spectroscopic analyses of gravity-sensitive line strengths give growing evidence towards an excess of low-mass stars in massive early-type galaxies (ETGs). Such a scenario requires a bottom-heavy initial mass function (IMF). However, strong constraints can be imposed if we take into account galactic chemical enrichment. We extend the analysis of Weidner et al. and consider the functional form of bottom-heavy IMFs used in recent works, where the high-mass end slope is kept fixed to the Salpeter value, and a free parameter is introduced to describe the slope at stellar masses below some pivot mass scale (M < MP = 0.5 M⊙). We find that no such time-independent parametrization is capable to reproduce the full set of constraints in the stellar populations of massive ETGs - resting on the assumption that the analysis of gravity-sensitive line strengths leads to a mass fraction at birth in stars with mass M < 0.5 M⊙ above 60 per cent. Most notably, the large amount of metal-poor gas locked in low-mass stars during the early, strong phases of star formation results in average stellar metallicities [M/H] ≲ -0.6, well below the solar value. The conclusions are unchanged if either the low-mass end cutoff, or the pivot mass are left as free parameters, strengthening the case for a time-dependent IMF.