  The nature of the GeV gamma-ray Galactic center excess (GCE) in the data of
Fermi-Large Area Telescope (LAT) is still to be unveiled. We present a new
analysis of the inner Galaxy Fermi-LAT data at energies above 10 GeV, based on
an innovative method which combines the skyFACT adaptive template fitting with
and the 1pPDF pixel-count statistics. We find a strong evidence for the GCE
also at high energies, $\sigma > 5$ regardless of the GCE spatial template.
Remarkably, our fits prefer the bulge morphological model over the dark matter
one at high significance, and show no evidence for an additional dark matter
template on top of the bulge component. Through the 1pPDF analysis, we find
that the model best describing the gamma-ray data requires a smooth, diffuse
GCE following a bulge morphology, together with sub-threshold point sources.
The 1pPDF fit reconstructs a consistent population of faint point sources down
at least to $10^{-12}$ ph cm$^{-2}$ s$^{-1}$. Between $10^{-12}$ ph cm$^{-2}$
s$^{-1}$ and $10^{-11}$ ph cm$^{-2}$ s$^{-1}$ the 1pPDF measures a number of
point sources significantly higher than the ones in the Fermi 4FGL catalog. The
robustness of our results brings further support to the attempt of explaining,
at least partially, the high-energy tail of the GCE in terms of a population of
point sources, likely corresponding to millisecond pulsars.
