Lol.
Just verified again. I've got some work to wrap up then I'll post a video.
Ok, just to clarify the parameters since I didn't make it clear before, you need to post a video long enough to show that it is continually charging, at least one to two hours in length. High quality need not apply, just enough to see that it is an iPad 3 and enough to see the plug icon.
As Apple admits, the battery will charge past the 100% point which is why the plug icon will show up when you power it up from a full charge, but within 13-35 minutes (it varies, oddly enough even if running the same tasks) the plug icon goes away, the charging icon comes on in its place, and then the battery declines. So showing a two or three minute video won't confirm anything at all. It has to be of substantial length. This is why I believe your video will confirm exactly what I am saying. The iPhone and iPad 2, on the other hand, will show the plug icon indefinitely because the units don't use more power than the charger is capable of outputting.
Make sure you show yourself setting the brightness at 100%, auto off. Wifi only is fine, although when plugged in it should obviously make no difference. I found the charging threshold to be somewhere around 65-70% brightness on my unit. Anything above that and it depletes. I don't do any processor-intensive tasks, unless one considers having iMessage as the only app open a processor-intensive task.
This is not just an issue with units purchased from my local store.
My first unit was a white 32GB AT&T model that was shipped from Apple.
The second and third units were/are black 32GB Verizon models from my local store.
Two models were made at one Foxconn facility, one was made at another.
The first unit had uneven color.
The last unit had a yellowing screen issue in one corner and poor off angle color hue.
This current unit has a perfect display.
People on Apple forums and two people I know personally that own the new iPads (one is WiFi 32GB white, one is Verizon 16GB black), one from here in Los Angeles, one from Oklahoma are having the same problems.
The advisor at Apple has encouraged me to troubleshoot, which is what I and others are trying to do. Any help anyone can offer is appreciated and please post your serial numbers or PM them to me. This way I can give them to Apple so they can research why your iPad is good or bad.