  An excess of $\gamma$ rays in the data measured by the Fermi Large Area
Telescope in the direction of the Galactic center has been reported in several
publications. This excess, labeled as the Galactic center excess (GCE), is
detected analyzing the data with different interstellar emission models, point
source catalogs and analysis techniques. The characteristics of the GCE,
recently measured with unprecedented precision, are all compatible with dark
matter particles (DM) annihilating in the main halo of our Galaxy, even if
other interpretations are still not excluded. We investigate the DM candidates
that fit the observed GCE spectrum and spatial morphology. We assume a simple
scenario with DM annihilating into a single channel but we inspect also more
complicated models with two and three channels. We perform a search for a
$\gamma$-ray flux from a list of 48 Milky Way dwarf spheroidal galaxies (dSphs)
using state-of-the-art estimation of the DM density in these objects. Since we
do not find any significant signal from the dSphs, we put upper limits on the
annihilation cross section that result to be compatible with the DM candidate
that fits the GCE. However, we find that the GCE DM signal is excluded by the
AMS-02 $\bar{p}$ flux data for all hadronic and semi-hadronic annihilation
channels unless the vertical size of the diffusion halo is smaller than 2 kpc
-- which is in tension with radioactive cosmic ray fluxes and radio data.
Furthermore, AMS-02 $e^+$ data rule out pure or mixed channels with a component
of $e^+ e^-$. The only DM candidate that fits the GCE spectrum and is
compatible with constraints obtained with the combined dSphs analysis and the
AMS-02 $\bar{p}$ and $e^+$ data annihilates purely into $\mu^+\mu^-$, has a
mass of 60 GeV and roughly a thermal cross section.
