  Context. The [Mg/Fe] abundance ratios are a fundamental fossil signature to
trace the chemical evolution of the disc. Despite of the huge observational and
theoretical effort, discrepancies between models and data are still present and
several explanations have been put forward to explain the [$\alpha$/Fe]
bimodality. Aims. In this work, we take advantage of a new AMBRE:HARPS dataset,
which provides new and more precise [Mg/Fe] estimations, as well as reliable
stellar ages for a subsample of stars, to study the evolution of the solar
neighbourhood. Methods. The above data are compared with detailed chemical
evolution models for the Milky Way, exploring the most used prescriptions for
stellar yields and different formation scenarios for the Galactic disc, i.e.
the delayed two-infall and the parallel model, also including prescriptions for
stellar radial migration. Results. We see that most of the stellar yields
struggle to reproduce the observed trend of the data and that semi-empirical
yields are still the best to describe the [Mg/Fe] evolution in the thick and
thin discs. In particular, most of the yields still predict a steeper decrease
of the [Mg/Fe] ratio at high metallicity than what is shown by the data. The
bulk of the data are well reproduced by the parallel and two-infall scenarios,
but both scenarios have problems in explaining the most metal-rich and
metal-poor tails of the low-$\alpha$ data. These tails can be explained in
light of radial migration from inner and outer disc regions, respectively.
Conclusions. Despite of the evidence of stellar migration, it is difficult to
estimate the actual contribution of stars from other parts of the disc to the
solar vicinity. However, the comparison between data and models suggests that
peculiar histories of star formation, such as that of the two-infall model, are
still needed to reproduce the observed distribution of stars.
