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tokens: γ-ray telescopes, and, possibly, by some of the currently operating ones. To our knowledge, these results have never been reported before. We have discussed how different choice of parameters (such as source redshift, primary proton energy and angular distribution, extragalactic magnetic field, and extragalactic background light models) would change the observable spectra. We have also provided a qualitative discussion of the main stages of cascade development and the effects of additional processes. This study could significantly facilitate future axion-like particle searches in the optically thick region of blazar spectra. In addition, our work might be useful in extreme TeV blazar studies as well as in several related branches of γ-ray astronomy, including extragalactic background light and extragalactic magnetic field measurements. APPENDIX A: THE IMPACT OF FLUCTUATIONS ON THE OBSERVABLE SPECTRA The observable spectrum in the framework of the MHCM depends on the EGMF structure, in particular, on the number of proton trajectories used in our study. In Fig. A1 , we show the observable SEDs averaged over various numbers of proton trajectories. We conclude that the statistics of 10 2 trajectories is sufficient to obtain a stable enough average observable spectrum for E 10 TeV, and the use of 10 3 trajectories already allows one to robustly determine the shape of the spectrum for E 30 TeV. Figure A1. Average observable SEDs for the KD10 EBL model option and the following numbers of trajectories: 10 (black curve), 100 (red curve), 10 3 (green curve), and 10 4 (blue curve). In addition, we have calculated the observable SEDs for 10 2 individual proton trajectories to demonstrate the impact of fluctuations on the shape of the observable spectrum (see Fig. A2 ). The blue-to-red gradient of colours gives a visual representation to the values of the ratios of different individual SEDs at two benchmark energies: 10 TeV and 100 GeV. Blue colours represent lower ratios, while red colours – higher ones. We note that at E = 5 TeV, the maximal difference between the flux corresponding to 95 per cent of the trajectories is limited to the factor of 2.5. Figure A2. Observable SEDs for the KD10 EBL model option and 10 2 individual trajectories; colours represent the ratios of individual SEDs at 10 TeV and 100 GeV. APPENDIX B: DISTRIBUTION OF THE CRITICAL DISTANCE In Fig. B1 , we present a histogram of a distribution on ‘the critical distance’ L cr (see Section 4.1 for definition of this parameter). The normalization of this histogram corresponds to the case of discrete probability density function, i.e. with summation over the bins of the histogram, with the N th bin containing the largest value of L cr . Figure B1. Histogram of the critical distance L cr distribution (red curve). APPENDIX C: SPECTRA WITH AND WITHOUT TPP, URB, and DPP In Fig. C1 , we present histograms of the observable spectra of γ-rays and electrons calculated with the CRPropa3 code for