Haha. You have to put a Bronze Bed in your Mog House, and set it up in Lay out.. then wait for the next Conquest Tally, and your Moogle should ask you for 2 items (cheap ones) when you click on him. Then you trade those 2 items to him, and he will expand your Moogle Safe to 60.. For the next quests, you need to set up a Mahogany bed in your MH.. and for the rest, check
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