I had no involvement in the paper, though I did read an early draft of it.
The consensus for the origin of these radiocarbon excess events is one (or more) Solar Proton Events. See for example Usoskin et al. (2013). It had been conjectured that a close gamma ray burst or supernova may have been the cause, but such events do not appear to have the correct radiation spectrum to generate the measured isotopic anomalies in 14C, 10Be and 36Cl (see Mikhaldi, 2014 for example).