  For more than two years, we monitored with the HARPS-N spectrograph the 400
Myr-old star HD\,63433, which hosts two close-in (orbital periods $P_b\sim7.1$
and $P_c\sim20.5$ days) sub-Neptunes detected by the TESS space telescope, and
it was announced in 2020. Using radial velocities and additional TESS
photometry, we aim to provide the first measurement of their masses, improve
the measure of their size and orbital parameters, and study the evolution of
the atmospheric mass-loss rate due to photoevaporation. We tested
state-of-the-art analysis techniques and different models to mitigate the
dominant signals due to stellar activity that are detected in the radial
velocity time series. We used a hydro-based analytical description of the
atmospheric mass-loss rate, coupled with a core-envelope model and stellar
evolutionary tracks, to study the past and future evolution of the planetary
masses and radii. We derived new measurements of the planetary orbital periods
and radii ($P_b=7.10794\pm0.000009$ d, $r_b=2.02^{+0.06}_{-0.05}$ $R_{\oplus}$;
$P_c=20.54379\pm0.00002$ d, $r_c=2.44\pm0.07$ $R_{\oplus}$), and determined
mass upper limits ($m_b\lesssim$11 $M_{\oplus}$; $m_c\lesssim$31 $M_{\oplus}$;
95$\%$ confidence level), with evidence at a 2.1--2.7$\sigma$ significance
level that HD\,63433\,c might be a dense mini-Neptune with a Neptune-like mass.
For a grid of test masses below our derived dynamical upper limits, we found
that HD\,63433\,b has very likely lost any gaseous H-He envelope, supporting
HST-based observations that are indicative of there being no ongoing
atmospheric evaporation. HD\,63433\,c will keep evaporating over the next
$\sim$5 Gyr if its current mass is $m_c\lesssim$15 $M_{\oplus}$, while it
should be hydrodynamically stable for higher masses.
