Sad I know, but I thought it would be fun to work it out:
Height of a badminton net is 1.55m at the posts (rule 1.10), so let us say shuttle is 1.60m from the ground. Half-length of court is 6.7m, so the angle is the arc-tangent of the ratio, which comes out at 13.43 degrees. However, the diagonal of a half-court (Pythagoras of 6.7 and 6.1m) is 9.061m, which similarly gives an angle of 10.01 degrees.
Taking the smallest skirt diameter of 58mm (rule 2.2.3), the longest feather length of 70mm (2.2.2) and the largest base diameter of 28mm (2.2.5) (via the arc-sine of the ratio of difference in radii to the length of the feather), I get a half-angle of 12.37 degrees. This is greater than the straight drive angle but not the cross.
Similarly, taking the average of the values quoted in the the 3 shuttle rules, I get 16.05 degrees, which is greater than even the straight drive. Finally, taking the most optimistic values, the half-angle is 20.29 degrees, which is well over both.
I would therefore argue that it is technically possible for a shuttle's feathers to land 'in' whilst the cork is 'out', albeit very unlikely.