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Is here any correlation between the values returned from Nikon's Auto AF Fine Tune and those values you need to enter via Sigma's dock (puck) for Sigma's Art and/or Sports lenses? 

There is little correlation of Nikon's auto AF fine tune to anything including correct focus unfortunately.... The numbers can swing wildly from test to test taken back to back. But to answer your question in a different way, 1 point of adjustment in Nikonese is close to but not exactly 1 point of adjustment in Sigmaese. So the way to do it is to draw a matrix for all of the adjustments that the Sigma dock allows for your lens, then measure each of those situations using Nikon's AFFT (manually, not the automatic method with a long ruler attached to a lens align II and the camera tethered to a computer so that you can make an accurate determination). Once the table is complete, enter all those values in the lens and reset the camera to zero offset. Now do it all again and adjust the values in the table by the amount of the new readings and enter those new values into the lens via the dock. That will get you very close. You can go for a third round if you want to really fine tune them some more.