Non-Congruent Tilings of the Plane - a bit more

. The basic question of whether one can tile the plane with mutually non-congruent triangles all of equal area and perimeter has been answered in the negative by Kupaavski, Pach and Tardos (as already noted in this earlier post here: )

Another question: If one looks for tiling the plane with non-congruent triangles of equal area and equal diameter (diameter of a triangle is its longest side) what happens? I the answer is in negative, can the plane be tiled with convex pieces of equal area and diameter with no constraint on the numbers of sides? And what about other combinations of quantities maintained constant over the mutually non-congruent tiles?

I don't know if an answer to these questions follows naturally from the above mentioned paper. A few variants of the basic question answered in the above-mentioned paper had been recorded in earlier posts here (especially this post) but if I am not mistaken, the "area & diameter constant" case wasn't mentioned.