  It has been recently demonstrated that both, a classical Schwarzschild black
hole (BH), and a dense concentration of self-gravitating fermionic dark matter
(DM) placed at the Galaxy centre, can explain the precise astrometric data
(positions and radial velocities) of the S-stars orbiting SgrA*. This result
encompasses the 17 best resolved S-stars, and includes the test of general
relativistic effects such as the gravitational redshift in the S2-star. In
addition, the DM model features another remarkable result: the dense core of
fermions is the central region of a continuous density distribution of DM whose
diluted halo explains the Galactic rotation curve. In this Letter, we
complement the above findings by analyzing in both models the relativistic
periapsis precession of the S2-star orbit. While the Schwarzschild BH scenario
predicts a unique prograde precession for S2, in the DM scenario it can be
either retrograde or prograde, depending on the amount of DM mass enclosed
within the S2 orbit, which in turn is a function of the DM fermion mass. We
show that all the current and publicly available data of S2 can not
discriminate between the two models, but upcoming S2 astrometry close to next
apocentre passage could potentially establish if SgrA* is governed by a
classical BH or by a quantum DM system.
