Are LGRBs biased tracers of star formation? Clues from the host galaxies of the $Swift$/BAT6 complete sample of bright LGRBs III: Stellar masses, star formation rates and metallicities at $z>1$

(Abridged) Long gamma-ray bursts (LGRB) have been suggested as promising tracers of star formation owing to their association with the core-collapse of massive stars. The goal of this work is to characterise the population of host galaxies of LGRBs at 1 0.7 Zsun. Under this hypothesis we can expect LGRBs to trace star formation at z > 3, once the bulk of the star forming galaxy population are characterised by metallicities below this limit. The moderately high metallicity threshold found is in agreement with the conditions necessary to rapidly produce a fast-rotating Wolf-Rayet star a in close binary system, and could be accommodated by single star models under chemically homogeneous mixing with very rapid rotation and weak magnetic coupling.