We add a little to Tiling the plane with pairwise non-congruent rational triangles

Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "yes", can such an equal area tiling be achieved with the perimeters of the triangles upper bounded?

I couldn't figure out if the theorem(s) proved in https://web.archive.org/web/20210414080312/https://math.dartmouth.edu/~carlp/PDF/paper14.pdf would settle the above question.

Further question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same perimeter?