Non-Congruent Tiling - 11

This has been a lengthy series - mainly due to experts having produced a series of results on the basic theme of tiling the plane with mutually non-congruent tiles which nevertheless share common properties. The common properties intensively studied are area and perimeter. The latest results (known to me) are in https://www.sciencedirect.com/science/article/abs/pii/S0097316521000601 ("the Euclidean plane can be tiled by mutually non-congruent convex quadrangles all of same area and perimeter").

Before putting up the main content of this post, let me put links to earlier instalments in this series (there could be some overlap in content!):
- Parts 1 to 6
- Part 7
- Part 8
- Part 9
- Part 10

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Let me restate a Broad Question that was put up in some form or other earlier on these pages: Apart from {area, perimeter}, what could one say if the mutually non-congruent tiles have to share, say, {area, diameter} or {area, least width} or any such pair of global properties?

And one can ask if in addition to area and perimeter if one more quantity, say, diameter or least width needs to be equal among the non-congruent tiles, can the plane be tiled with quadrangles. Another possibility is: all non-congruent pieces ought to have some specified centralness coefficient (its value ranges from 1/2 for all triangles to 1 for centrally symmetric convex regions)
The following is still more speculative:

Moving on from tiling to covering and packing, one think of covering or packing the plane with a set of mutually non-congruent polygons with shared properties. Are there global results such as "if pieces have to be mutually non-congruent with same area and perimeter, the resulting packing/covering is always inferior to the best packing/covering with congruent copies of any tile that has the same area and perimeter."

And let me repeat, all of the above questions can be transplanted onto the hyperbolic plane! -