For about a decade the IceCube Neutrino Observatory has been observing a high-energy diffuse astrophysical neutrino flux. At these energies, an important source of background are the prompt atmospheric neutrinos produced in decays of charmed mesons that are part of cosmic-ray-induced air showers. The production yield of charmed mesons in the very forward phase space of hadronic interactions, and thus the flux of prompt neutrinos, is not well known and has not yet been observed by IceCube. A measurement of the flux of prompt neutrinos will improve the modeling of hadronic interactions in cosmic-ray induced air showers at high energies. Additionally, in the context of astrophysical neutrino measurements, understanding this background flux will improve the measurement precision of the spectral shape in the future. In particular, the analysis of up-going muon neutrino-induced tracks in IceCube provides a large sample of atmospheric neutrinos which likely includes prompt neutrinos. However, the measurement of a subdominant prompt neutrino flux strongly depends on the hypothesis for the dominant astrophysical neutrino flux. This makes the estimation of upper limits on the prompt neutrino flux challenging. We discuss the extent of this model dependency on the astrophysical flux and propose a method to calculate robust upper limits. Furthermore, a possible dedicated search of the prompt neutrino flux using multiple IceCube detection channels is outlined.