Non-congruent tiling - 17

A pointer to the previous instalment of this series.

We consider tiling with equal area, mutually non-congruent tiles:

Question: Can the plane be tiled with equal area triangles all of which are pair-wise non-congruent and with every edge in the layout having unique length?

Remark 1: If we require the pair-wise non-congruent equal area triangles all to have every angle unique, it seems possible based on the arguments given in https://arxiv.org/pdf/1603.09132.pdf.

Further question: The above question (both unique angles and unique edges variants) can be asked with 'perimeter' replacing 'area'. Note: It has been proved that the plane cannot be tiled by non-congruent triangles all with same area and same perimeter (https://arxiv.org/abs/1711.04504).

Here is the mathoverflow page with above question
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We now move to the hyperbolic plane:

1. Can the hyperbolic plane be tiled by mutually non-congruent equal area triangles?

2. Can the hyperbolic plane be tiled by mutually non-congruent equal area triangles with the further constraint that every angle is unique?