Remarks on atmospheric effect of D-foam in light of muon puzzle

Abstract: In our paper arXiv:2509.00552, we used a stringy model for quantum space-time foam to suggest that the so-induced subluminal Lorentz violation~(LV) for photons would not lead to experimentally unacceptable changes in the developments of particle showers initiated by cosmic $γ$-rays in the Earth's atmosphere, in contrast to other approaches to LV. The result indicated, nonetheless, at the same time that the foam can mildly modify the electromagnetic cascades under certain conditions, by suppressing pair creation on nuclei by primary photons. In this addendum, we consider how this modification affects the detection of extensive air shower~(EAS) initiated by an ultrahigh-energy cosmic-ray particle~(viz., a primary hadron), like proton with $E \sim 10<sup>{19}~\textrm{eV}$, given that secondary photon subshowers following $π<sup>{0}$ decays could be similarly influenced. We argue that fewer electrons would reach the detector and hence the energy of the primary particle may be underestimated due to foam effects, enhancing in such a way the muon content in EASs. This opens up the possibility of interpreting the alleged ``excess'' of muons, as reported by Auger and Telescope Array collaborations recently and many other experiments on high-energy cosmic rays, with a quantum-gravitational effect. Future observations are anticipated to confirm whether this anomaly really exists.