On Triangulations of the Plane

Back in 2014, I had asked here if the plane can be tiled by pairwise noncongruent triangles all of same area and perimeter and that question has been answered in the negative recently (https://arxiv.org/abs/1711.04504) and further aspects explored in for example, https://arxiv.org/abs/1905.08144. Now, let me record another line of thinking...

Basic question: Can the plane be divided into a vertex to vertex arrangement of non-overlapping triangles such that every edge has a unique rational length r that lies between 1 and some specific rational greater than 1?

If this is possible, one can apply further constraints such as "all triangles should have equal area (OR equal perimeter)". Alternatively, one can relax the vertex-to-vertex requirement to frame another question.

Note 1: Requiring the lengths of all edges to be integers rather than rationals would lead to the lengths of the triangles being unbounded even if a triangulation with all edges having unique lengths is possible (not sure if this is possible).

Here is a discussion page on this subject:
https://mathoverflow.net/questions/351047/triangulating-the-plane-using-edges-of-unique-rational-lengths