Non-congruent Tilings - 21

Although plenty has been written on this thread, a pretty obvious question seems to have escaped one's attention (at least, looking through all the earlier posts one didn't spot any trace thereof!). Here goes.
The main theorem proved by Kupaavski, Pach and Tardos on the basic question:

Theorem: There is no tiling of the plane with pairwise noncongruent triangles all of the same area and the same perimeter

Question: Can the plane be tiled by triangles all of same area and perimeter that are pair-wise non-congruent except that exactly 2 of them are mutually congruent? Of course, '2' can be generalized to 'finitely many mutually congruent triangles' or 'finitely many pairs of congruent triangles' etc.

This has been put up at mathoverflow: https://mathoverflow.net/questions/469784/trying-to-extend-a-theorem-on-tiling-with-mutually-non-congruent-triangles