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With the D3 AF even with the 70-200mm f2.8 VR II lens and the TC-20E III was not possible. The camera would not even hunt and it was manual focus most of the time. It was OK for bright sunlight with large high contrast subjects as with sports photography but poor for wildlife photography.
That's interesting. That combo makes an f/5.6 lens, like the 80-400 VR. It should work fine.
I just bought a used TC-20EII, not III, to use on my 70-200 VR I. It arrived a few days ago and I put it on the lens and tested it with my d3s and d7000. The tests were just shots of my dogs and other stuff around my yard, in all kinds of light from deep shade to bright sunlight. I didn't notice any problems with AF. Indeed, the combo worked better/faster than the 80-400, which is the reason I bought the TC. I would have thought that your newer versions would work at least as well.