If this matters to anyone, you CAN load cores into Training mode. Just tested it with a 5X HP core half an hour ago, and I was receiving 10k-ish exp, clean. But unless you're a massive AOE mobber e.g. DS or Lumi, and/or you have all the time in the world to do the ridiculous grinding and farming, I don't advise you to do that.
Hadriel
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