There exists a growing evidence for the anomalous transparency of the Universe for energetic gamma rays. Popular explanations include conversion of photons into hypothetical axion-like particles (ALPs) and back in astrophysical magnetic fields. Two distinctive scenarios of this conversion have been put forward: either it happens in the (host galaxy of the) gamma-ray source and in the Milky Way, or the photon-ALP oscillations take place in the intergalactic magnetic fields all along the way between the source and the observer. Here we point out that, given recent astrophysical constraints on ALPs and on intergalactic magnetic fields, these two mechanisms imply very different ALP masses and couplings. Therefore, confirmation of the anomalies and identification of one of the scenarios would mean cornering of ALP parameters to a particular narrow region. We discuss approaches to distinguish between the two mechanisms and present some preliminary indications in favour of the galactic scenario.