Yes, within 5 folds, the bear imprint can be made.
The approach taken here is to draw the bear design on the paper and start folding in symmetrical parts to match all the points onto one another.
1.Fold the paper in half for symmetry. (fold-1)
all other folds will be done on the folded paper.
2.align the most bottom point and the second bottom point and make a fold in the middle. this is the 2 holes in the bottom. (fold-2)
only one point would be remaining at the bottom.
3.Fold the top trapezium in half. through the normal of the parallel lines. (fold-3)
4.Fold such that the two asymmetrical points of the trapezium match. (fold-4).
only one fold point would remain in the top section
5. Fold such that the bottom fold point matched the top fold point. (fold-5)
6. Punch a hole and unfold the paper completely to get the bear shape.
Note: fold point is the approximate location of the point where we punch as to get the image.
Also, all folds are done by hand, by matching the edges and an approximate location of the fold points, there could be a very precise and accurate way of doing this too, with angles and lengths.
This is a quick way of getting a rough Bear Imprint.