  The system of two transiting Neptune-sized planets around the bright, young
M-dwarf AU Mic provides a unique opportunity to test models of planet
formation, early evolution, and star-planet interaction. However, the intense
magnetic activity of the host star makes measuring the masses of the planets
via the radial velocity (RV) method very challenging. We report on a 1-year,
intensive monitoring campaign of the system using 91 observations with the
HARPS spectrograph, allowing for detailed modelling of the $\sim 600\,
\rm{m\,s^{-1}}$ peak-to-peak activity-induced RV variations. We used a
multidimensional Gaussian Process framework to model these and the planetary
signals simultaneously. We detect the latter with semi-amplitudes of $\rm{K_b}
= 5.8 \pm 2.5\, \rm{m\,s^{-1}}$ and $\rm{K_c} = 8.5 \pm 2.5\, \rm{m\, s^{-1}}$,
respectively. The resulting mass estimates, $\rm{M_b} = 11.7 \pm 5.0\,
\rm{M_{\rm \oplus}}$ and $\rm{M_c} = 22.2 \pm 6.7\, \rm{M_{\rm \oplus}}$,
suggest that planet b might be less dense, and planet c considerably denser
than previously thought. These results are in tension with the current standard
models of core-accretion. They suggest that both planets accreted a H/He
envelope that is smaller than expected, and the trend between the two planets'
envelope fractions is the opposite of what is predicted by theory.
