In the last decade, growing evidence has emerged supporting a non-universal stellar initial mass function (IMF) in massive galaxies, with a larger number of dwarf stars with respect to the Milky Way (bottom-heavy IMF). However, a consensus about the mechanisms that cause IMF variations is yet to be reached. Recently, it has been suggested that stars formed early-on in cosmic time, via a star formation burst, could be characterized by a bottom-heavy IMF. A promising way to confirm this is to use relics, ultra-compact massive galaxies, almost entirely composed by these 'pristine' stars. The INvestigating Stellar Population In RElics (INSPIRE) Project aims at assembling a large sample of confirmed relics, that can serve as laboratory to investigate on the conditions of star formation in the first 1-3 Gyr of the Universe. In this third INSPIRE paper, we build a high signal-to-noise spectrum from five relics, and one from five galaxies with similar sizes, masses, and kinematical properties, but characterized by a more extended star formation history (non-relics). Our detailed stellar population analysis suggests a systematically bottom-heavier IMF slope for relics than for non-relics, adding new observational evidence for the non-universality of the IMF at various redshifts and further supporting the above proposed physical scenario.