Stretching Fair Partitions - 2

We add another very speculative claim to those in the last post.

"If a convex planar region C has the property for any positive integer n that at least one convex fair partition of C into n convex pieces has at least 2 (or maybe 3, say) of the pieces mutually congruent, then for any n, C allows a convex fair partition into n pieces that are all mutually congruent - and further, C is either a sector of a disk or parallelogram."