We add a bit to Tiling the plane with pairwise non-congruent rational triangles. The solutions given there show tilings of the plane with pairwise non-congruent rational triangles that are either (1) unbounded in size or (2) have all tiles sharing at least one edge length.
Question: Can we build a tiling of the plane with pair-wise non-congruent rational triangles such that the size of the tiles is bounded (both above and below) and no finite fraction of the total set of edge lengths is equal (extreme cases would be the set of edge lengths being all different or an edge length shared between at most an adjacent pair of tiles)?
