Results of our model for a LGRB efficiency, as defined in Eq. (C.1), with Z_{th, low} = 8.0, Z_{th, up} = 8.6, and η₀ set at $5\cdot {10}^{-5}{M}_{\odot }^{-1}$. The figure is similar to the right part of Fig. 6, where the blue distributions show the R_LGRB model predictions and are compared to the results of Graham et al. (2023), Perley et al. (2016c), and Ghirlanda & Salvaterra (2022). The integrated metallicity (for z ≤ 2.5) and host galaxy mass (for z ≤ 0.5) distributions are compared respectively to the results of Graham et al. (2023, through the method of Curti et al. 2017, 2020) and the local LGRB sample listed in Appendix A. We note that panel C shows the SFR and LGRB rates on two different scales (cf. panels C and F from Fig. 6).