Non-congruent tiling - 14

Here is the previous part of this series:
Revisiting a question posed in the first instalment of this series, back in 2014 (https://nandacumar.blogspot.com/2014/12/filling-plane-with-non-congruent-pieces.html)

In https://arxiv.org/pdf/1905.08144.pdf ?Frettloh and Richter prove: "There is a vertex-to-vertex tiling of the plane by pairwise incongruent triangles of unit area and uniformly bounded perimeter. "

Remark: Their construction seems to achieve a tiling with incongruent equal area triangles whose perimeters lie within an arbitrarily small range (equal is not possible).

Question: Will a similar construction achieve a tiling by pairwise incongruent triangles of unit perimeter and bounded area (area bounded both above and below) with/without the v-to-v property?

Let me also add another recent query posted on mathoverflow: https://mathoverflow.net/questions/421333/on-rigid-packings-of-the-plane-with-a-constraint