When you send the books to a Kobo ereader, the On device green check lets you know the books have been sent. When you reconnect, the green check is generated from the database contents. If the books have not been imported on the Kobo to generate the database entries, they will not show as on device. The import screen is basically a black screen with a progress bar that goes from 6% to 94% as the books are imported.
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