A guy named Peter Beckett did some testing of that theory, there's a thread on Nikonians
http://www.nikonians.org/forums/dcbo...id=87491#87496
You may need membership to see that thread, not certain (and apologies if so).
He didn't have a D4 to experiment with, but did some experiments with a D800 and different power sources, and did timings with a scope (i.e. real scientific approach, not just "feels the same" like me), with the goal to see how power available affected focus speed, with a scope doing the timing very precisely.
I hesitate to abbreviate his findings as there was some ambiguity in there, as it's not clear cut. But if you are interested it might make interesting reading.