Experimental tests of sub-surface reflectors as an explanation for the ANITA anomalous events

The balloon-borne ANITA experiment is designed to detect ultra-high energy neutrinos via radio emissions produced by an in-ice shower. Although initially purposed for interactions within the Antarctic ice sheet, ANITA also demonstrated the ability to self-trigger on radio emissions from ultra-high energy charged cosmic rays interacting in the Earth's atmosphere. For showers produced above the Antarctic ice sheet, reflection of the down-coming radio signals at the Antarctic surface should result in a polarity inversion prior to subsequent observation at the $\sim$35-40 km altitude ANITA gondola. ANITA has published two anomalous instances of upcoming cosmic-rays with measured polarity opposite the remaining sample of $\sim$50 UHECR signals. The steep observed upwards incidence angles (25--30 degrees relative to the horizontal) require non-Standard Model physics if these events are due to in-ice neutrino interactions, as the Standard Model cross-section would otherwise prohibit neutrinos from penetrating the long required chord of Earth. Shoemaker et al. posit that glaciological effects may explain the steep observed anomalous events. We herein consider the scenarios offered by Shoemaker et al. and find them to be disfavored by extant ANITA and HiCal experimental data. We note that the recent report of four additional near-horizon anomalous ANITA-4 events, at $>3\sigma$ significance, are incompatible with their model, which requires significant signal transmission into the ice.