I was able to get the "official" HDMI dongle working on the Galaxy Tab 3 8.0 by ensuring the amperage of a 5.0V USB wall charger was 1.2A or greater. It did not work (got loop of disconnect/connect message as OP observed) with chargers below 1.0A. A generic 1.2A charger and Apple iPad Mini 2.1A...
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