This could be a grounding issue, but not a typical ground loop. Adding a ground connection from GT to Axe-Fx seems to largely fix the problem. The USB connection does that normally but if you're not always be using USB, adding a jumper inside the GT to ground the MIDI input jack seems work.
According to the MIDI standard, MIDI outputs are supposed to be grounded, but MIDI inputs are not, in order to prevent ground loops. However, in this particular case it can help. It's easy to remove the jumper if installed if it ever becomes necessary - just remove the bottom lid, find the wire between the MIDI jacks and cut it with some wire cutters.