Remember that one of the basic ideas of the Authagraph map is that you can grab different views of the world without (heavy) interruptions or repetitions simply by moving or rotating a rectangle, as is shown here:
I dont’t think that this is possible when you put the south pole to the tip of the tetrahedron. Because then, North America and Eurasia will always form a "circle" (well, something like a circle anyway) around the north pole which makes it hard to cut a full world map in rectangular form.
To illustrate what I’m talking about, here's an image of another tetrahedral projection (van Leeuwen), centered to the north pole:
Try to tesselate that, you’ll see what I mean. Even when you move the center a bit, I don’t think that you’ll ever come up with a solution that can be used as mentioned above.