  The $\textit{Fermi}$-LAT Collaboration recently presented a new catalog of
gamma-ray sources located within the $40^{\circ} \times 40^{\circ}$ region
around the Galactic Center~(Ajello et al. 2017) -- the Second Fermi Inner
Galaxy (2FIG) catalog. Utilizing this catalog, they analyzed models for the
spatial distribution and luminosity function of sources with a pulsar-like
gamma-ray spectrum. Ajello et al. 2017 v1 also claimed to detect, in addition
to a disk-like population of pulsar-like sources, an approximately 7$\sigma$
preference for an additional centrally concentrated population of pulsar-like
sources, which they referred to as a "Galactic Bulge" population. Such a
population would be of great interest, as it would support a pulsar
interpretation of the gamma-ray excess that has long been observed in this
region. In an effort to further explore the implications of this new source
catalog, we attempted to reproduce the results presented by the
$\textit{Fermi}$-LAT Collaboration, but failed to do so. Mimicking as closely
as possible the analysis techniques undertaken in Ajello et al. 2017, we
instead find that our likelihood analysis favors a very different spatial
distribution and luminosity function for these sources. Most notably, our
results do not exhibit a strong preference for a "Galactic Bulge" population of
pulsars. Furthermore, we find that masking the regions immediately surrounding
each of the 2FIG pulsar candidates does $\textit{not}$ significantly impact the
spectrum or intensity of the Galactic Center gamma-ray excess. Although these
results refute the claim of strong evidence for a centrally concentrated pulsar
population presented in Ajello et al. 2017, they neither rule out nor provide
support for the possibility that the Galactic Center excess is generated by a
population of low-luminosity and currently largely unobserved pulsars.
