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."

Stretching the Fair Partition question

Can one make claims of the following type?
"If some convex region C allows partition into n convex pieces all of equal area, perimeter and one more quantity, say diameter or least width for all values of n (or infinitely many values of n), then all pieces are necessarily congruent."

If the above is true, one can stretch things a bit and guess: "If all pieces are congruent for all n, C is necessarily a sector of a disk (with the full disk as a limiting case) or a parallelogram (including rectangles). If 'for all n' is relaxed to 'infinitely many values of n', one also has the case of C being a triangle." This latter guess was once posted at mathoverflow.

Note: perimeter and diameter can be nonzero even when a polygon is degenerate but not area or least width. Basically the question/claim is about 3 quantities being equal among pieces (with 2 of the quatities being like perimeter and one like area or vice versa). If 3 quantities being equal isnt enough for the congruence claim to hold, consider 4!