tokens: field for the IC emission, the pionic γ-ray emission in our model is not able to account for both the observed γ-ray luminosities and the spectral shapes of galaxies with low SFRs such as the SMC alone. As diffusive losses become more relevant in these low-density galaxies, this leads to a steepening of the CR proton spectra (depending on the exact energy dependence incorporated in the diffusion time-scale). This necessarily results in steeper pion-decay γ-ray spectra, which would be in conflict with the observations if this were the only relevant emission channel. Hence, a leptonic contribution to the γ-ray emission from IC scattering is indispensable in our models. One of the arguments made in the literature in order to conclude a subdominant leptonic contribution to the total γ-ray emission depends on several underlying assumptions that are only valid in the nuclear regions of starbursts, i.e. for high gas densities, magnetic field strengths, and wind velocities. The magnetic field strength is either deduced from an energy equipartition argument, or from fitting several free parameters in one-zone models (see e.g. Torres 2004 ; Lacki et al. 2010 , 2011 ; Yoast-Hull et al. 2013 ) or axisymmetric two-dimensional models (Buckman et al. 2020 ) in order to simultaneously match the observations. The former approach assumes that the kinetic energy density is equal to the magnetic energy density, . Assuming turbulent velocities of and gas densities of yields in this approach magnetic field strengths of the order of . The fast cooling of CR electrons due to synchrotron emission in these strong magnetic fields has been proposed to be one of the reasons for suppressing the possible leptonic contribution to the γ-ray emission (Lacki et al. 2011 ). In contrast, γ-ray observations of the Galactic Centre suggest that CRs might not be able to penetrate into the densest regions and thus may be far from calorimetry (Crocker et al. 2011a , b ). However, those extreme properties are at most reached in the very central region of our simulated galaxies. Additionally, the energy density of the interstellar radiation field (or its equivalent magnetic field strength, see Fig. 1 , upper panels) persists throughout the whole galaxy. The cooling of the electrons is hence not fully dominated by synchrotron cooling, enabling IC losses to be non-negligible, except in the very central regions of starburst galaxies. In particular, in the low-density ISM of dwarfs, we find that the IC emission might be able to contribute up to 40 per cent of the total γ-ray luminosity, as can be seen in Fig. 4 (middle panels). Still, our models of starburst galaxies M82 and NGC 253 (that are chosen to match the observed SFRs and L γ ) also match their observed radio luminosities at 1.4 GHz. We will address the radio emission of our simulated galaxies in Paper III in more detail. Our findings suggest that SF galaxies are not reaching the calorimetric limit, and depart from this limit even
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