Results of the atmospheric evolution modelling with MESA. Top panel: initial atmospheric mass fraction against the initial mass of the planet allowing us to reproduce the present-day parameters of TOI-421 b according to our atmospheric evolution models. The admissible combinations of $f_{\text{atm}}^{\text{start}}$ and M_pl,0 are shown by the shaded areas, and different colours correspond to different assumptions on the stellar rotation history (i.e. on the rotation period at 150 Myr), as indicated in the legend. Bottom panel: same as top, but for TOI-421 c.