Non-Congruent Tilings - 6

This post continues the 'broken chain' of posts on the non-congruent tiling front:

Here are links to the first 5 installments (am keeping them here for reference):

1. part 1
2. part 2
3. part 3
4. part 4
5. part 5
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Far more importantly, here are some of the major expert findings in this ongoing story (I am far behind the story!):

- The Euclidean 2D plane cannot be tiled with mutually noncongruent triangles all of the same area and perimeter. (Kupaavski, Pach, Tardos https://arxiv.org/pdf/1711.04504.pdf)
- non congruent tiling is possible if the triangles have equal area and bounded perimeter. (Frettloh https://arxiv.org/abs/1603.09132)
- the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. (Dirk Frettloh and Christian Richter https://arxiv.org/abs/2004.01034)

Remarks: I just gathered from an expert that the same question (or set of questions) could have very different behavior and answers on the Hyperbolic plane. One hopes experts would look into those possibilities.

The hyperbolic possibility was mentioned in post number 3 listed above (an update made in 2018). However, despite a lot of effort (not focussed but substantial), I am yet to gain sufficient command over Hyperbolic geometry to attack these questions.

In elliptic geometry, since there is roughly speaking less space around each point than hyperbolic or even Euclidean, one suspects non congruent tilings to be harder than in Euclidean plane. So, impossibility results may be easier to prove.