So I conquered it, and completed all fights and conversations. I'll try that fight thing, though. Will a training fight work?
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I don't think training fights will work, as I think I designed them to restore your health after you were done, but maybe..?
A guaranteed way to get it to work would be to switch your meals (via Hana in the kitchen) to "Caviar on Toast" which gives you a 50% chance of your territories being attacked by Org Goons.