Not yet reviewed by Pith; the record is open.

Many Beyond-Standard Model physics signatures are enhanced in high-energy neutrino interactions. To explore these signatures, ultra-large Cherenkov detectors such as IceCube exploit event samples with charged current muon neutrino interactions > 1 TeV. Most of these interactions occur below the detector volume, and produce muons that enter the detector. However, the large spacing between detectors leads to inefficiency for measuring muons with energies below or near the critical energy of 400 GeV. In response, IceCube has built a densely instrumented region within the larger detector. This provides large samples of well-reconstructed interactions that are contained within the densely instrumented region, extending up to energies of ~50 GeV. This leaves a gap of relatively unexplored atmospheric-neutrino events with energies between 50 GeV and 1 TeV in the ultra-large detectors. In this paper we point out that interesting Beyond Standard Model signatures may appear in this energy window, and that early running of the DUNE far detectors can give insight into new physics that may appear in this range.