The leading explanation of the $textit{Fermi}$ Galactic center $gamma$-ray excess is the extended emission from a unresolved population of millisecond pulsars (MSPs) in the Galactic bulge. Such a population would, along with the prompt $gamma$ rays, also inject large quantities of electrons/positrons ($e^pm$) into the interstellar medium. These $e^pm$ could potentially inverse-Compton (IC) scatter ambient photons into $gamma$ rays that fall within the sensitivity range of the upcoming Cherenkov Telescope Array (CTA). In this article, we examine the detection potential of CTA to this signature by making a realistic estimation of the systematic uncertainties on the Galactic diffuse emission model at TeV-scale $gamma$-ray energies. We forecast that, in the event that $e^pm$ injection spectra are harder than $E^{-2}$, CTA has the potential to robustly discover the IC signature of a putative Galactic bulge MSP population sufficient to explain the GCE for $e^pm$ injection efficiencies in the range $approx 2.9-74.1%$, or higher, depending on the level of mismodeling of the Galactic diffuse emission components. On the other hand, for spectra softer than $E^{-2.5}$, a reliable CTA detection would require an unphysically large $e^pm$ injection efficiency of $gtrsim 158%$. However, even this pessimistic conclusion may be avoided in the plausible event that MSP observational and/or modeling uncertainties can be reduced. We further find that, in the event that an IC signal were detected, CTA can successfully discriminate between an MSP and a dark matter origin for the radiating $e^pm$.
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