Not only is the inner band on Mklll and lll* rifles tapered, the hole is not actually round.
There are two little "flat" section in the upper half. These are aligned at 60 degrees to the horizontal. Good filing exercise for the apprentices!
I guess that this provided some sort of guided, centralising bearing, of a similar nature to the little extra window cut above the muzzle exit on the nose-cap.
On the road again: When I get home tomorrow I will see if I can kick the scanner into life and post the actual drawing.
Must also have a closer look at the small box of inner bands again!