Non-congruent Tiling - 13

Here is Part 12 of this series...

Good News: Dirk Frettloh and Christian Richter have written a preprint where they prove that the Euclidean plane can be dissected into mutually incongruent convex pentagons of the same area and same perimeter. Some further open questions are also stated. So more results are round the corner.

For this post, let me return to a question from part 7:

1. For what values of integer n can the surface of a sphere be partitioned into n convex and mutually non-congruent pieces of same area? (For closed surfaces, convexity could be viewed as geodesic convexity). If such a partition exists for some n, one can ask if one can achieve a fair convex partition - pieces with same area and same perimeter - for some of those n's.

Note: one can replace sphere by general ellipsoids or tori or polyhedra.

Speculation: Even a negative result like: "the spherical surface (or maybe even any ellipsoid or say, toroid) cannot be cut into any number of mutually non-congruent convex pieces or same area (or for that matter, 'same area and same perimeter')" would be interesting!
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Update(10th Feb, 2022): Here is a mathoverflow discussion page on the question. It has been shown there by Jukka Kohonen that if only area is to be equalized among n non-congruent pieces into which a spherical surface is to be cut, it can be done - indeed in a vertex to vertex fashion. And it seems achieving fair partition of the surface into n non-congruent pieces would be much harder due to the number of constraints.