Those are UHECRs (ultra-high energy cosmic rays), mostly likely protons, but perhaps nuclei (up to) iron - it's kinda difficult to determine the composition of these very rare beasts.

The highest gammas that I can recall being reported are ~20 TeV, from an innovative CR 'telescope' (whose name escapes me just now), which uses water instead of air.

When (if?) LOFAR is up and running, it should provide some nice independent checks on this.

However, I think the second part of the last sentence of the (pre-print) abstract is the most important: "but we cannot exclude the importance of some other source effect" - the source is a physical regime so far from anything probed so far in any Earthly lab (and likely to remain unprobed for possibly centuries to come) that constraining 'source effects' will likely be a headache for many decades yet.