Warranty issues aside and given my experience with OB shafts, I wouldn't turn them down at all. There's a 5/16" hollow that runs the length of the shaft.
If you're looking for a thinner OB, look into the OB2.
The OB-1 we like to keep above 12.25mm for the very same reason that KJ mentioned. The core is fairly large and we don't want the wall thickness to get too thin.
The OB-2, for the same reasons, should not go below 11mm.
As far as which one to choose, the OB-2 hits a little harder because the core is smaller than the one in the OB-1