The geodesic motion of S2 and G2 as a test of the fermionic dark matter nature of our galactic core

[Abridged] The S-stars motion around the Galactic center (Sgr A*) implies the existence of a compact source with a mass of about $4\times 10^6 M_\odot$, traditionally assumed to be a massive black hole (BH). Important for any model is the explanation of the multiyear, accurate astrometric data of S2 and the challenging G2: its post-pericenter velocity decelerates faster than expected from a Keplerian orbit around the putative BH. This has been reconciled in the literature by acting on G2 a drag force by an accretion flow. Alternatively, we show that the S2 and G2 motion is explained by the "core-halo" fermionic dark matter (DM) profile of the fully-relativistic Ruffini-Arg\"uelles-Rueda (RAR) model. It has been already shown that for 48-345 keV fermions, it accurately fits the rotation curves of the Milky-Way halo. We here show that, for a fermion mass of 56 keV, it explains the time-dependent data of the position (orbit) and light-of-sight radial velocity (redshift function $z$) of S2 and G2, the latter without a drag force. We find the RAR model fits better the data: the mean of reduced chi-squares of the orbit and $z$ data are, for S2, $\langle\bar{\chi}^2\rangle_{\rm S2, RAR}\approx 3.1$ and $\langle\bar{\chi}^2\rangle_{\rm S2, BH}\approx 3.3$ while, for G2, $\langle\bar{\chi}^2\rangle_{\rm G2, RAR}\approx 20$ and $\langle\bar{\chi}^2\rangle_{\rm G2, BH}\approx 41$. For S2 the fits of the $z$ data are comparable, $\bar{\chi}^2_{z,\rm RAR}\approx 1.28$ and $\bar{\chi}^2_{z,\rm BH}\approx 1.04$, for G2 only the RAR model fits, $\bar{\chi}^2_{z,\rm RAR}\approx 1.0$ and $\bar{\chi}^2_{z,\rm BH}\approx 26$. In addition, the critical mass for the gravitational collapse of a degenerate 56 keV-fermion DM core into a BH is $\sim 10^8 M_\odot$, which may be the initial seed for the formation of the observed central supermassive BH in active galaxies, such as M87.