Non-Congruent Tiling - 12

Link to Previous instalment of this series.

Question: Is there any non-right triangle T that can be tiled by finitely many mutually non-congruent triangles all similar to T?

I am aware that It was proved by Tutte that the equilateral triangle does not have the above property. On the other hand, every right triangle can be tiled by just 2 right triangles - mutually noncongruent and similar to it.

Basically, one would like to know if the result of https://www.math.nyu.edu/~pach/publications/equilateraltiling052218.pdf is equilateral specific.

Note: Even if a given triangle cannot be cut into finite number of self-similar and mutually non-congruent copies, one CANNOT rule out a tiling of the plane with non-congruent and similar copies of it.

Indeed, in Part 3 of this series, the question was asked:
Can the plane be tiled with triangles that are similar to one another but all of different sizes? Further we could insist on both upper and lower bounds on the sizes

Although a right triangle can be cut into 2 self similar and mutually non congruent pieces, I don't readily see how one can tile the plane with mutually similar and non-congruent right triangles - with bound on the sizes. Tiling with non-congruent squares can be done with a lower bound on the size of the squares - indeed, a 'fibonacci square spiral layout' can be formed and one of the two unit squares therein can be further 'carte blanched'. But this square spiral too has no upper bound on the size of tiles.