  The TOI-421 planetary system contains two sub-Neptune-type planets and is a
prime target to study the formation and evolution of planets and their
atmospheres. The inner planet is especially interesting as the existence of a
hydrogen-dominated atmosphere at its orbital separation cannot be explained by
current formation models without previous orbital migration. We jointly
analysed photometric data of three TESS sectors and six CHEOPS visits as well
as 156 radial velocity data points to retrieve improved planetary parameters.
We also searched for TTVs and modelled the interior structure of the planets.
Finally, we simulated the evolution of the primordial H-He atmospheres of the
planets using two different modelling frameworks. We determine the planetary
radii and masses of TOI-421 b and c to be $R_{\rm b} = 2.64 \pm 0.08 \,
R_{\oplus}$, $M_{\rm b} = 6.7 \pm 0.6 \, M_{\oplus}$, $R_{\rm c} = 5.09 \pm
0.07 \, R_{\oplus}$, and $M_{\rm c} = 14.1 \pm 1.4 \, M_{\oplus}$. We do not
detect any statistically significant TTV signals. Assuming the presence of a
hydrogen-dominated atmosphere, the interior structure modelling results in both
planets having extensive envelopes. While the modelling of the atmospheric
evolution predicts for TOI-421 b to have lost any primordial atmosphere that it
could have accreted at its current orbital position, TOI-421 c could have
started out with an initial atmospheric mass fraction somewhere between 10 and
35%. We conclude that the low observed mean density of TOI-421 b can only be
explained by either a bias in the measured planetary parameters (e.g. driven by
high-altitude clouds) and/or in the context of orbital migration. We also find
that the results of atmospheric evolution models are strongly dependent on the
employed planetary structure model.
