Non-Congruent Tiling - 10

The basic non-congruent question was "is it possible to tile the plane with pair-wise non-congruent triangles all of same area and same perimeter?"

Over a few papers, the question was answered in the negative some years ago.
https://arxiv.org/abs/1603.09132
https://www.sciencedirect.com/science/article/abs/pii/S0195669818300957

Now, one would like to ask again (following https://nandacumar.blogspot.com/2020/11/non-congruent-tilings-9.html) whether the plane can be tiled with mutually non-congruent triangles with
1. same area and diameter
2. same perimeter and diameter.

Note: Perhaps one can replace diameter with 'least width' to generate more questions.
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Another possible variant to the non-congruent tiling problem is tiling the plane with:
- mutually non-congruent isosceles triangles (already introduced in http://nandacumar.blogspot.com/2020/06/non-congruent-tilings-8.html).
Now, we note some subvariants to this question:(1) isoceles triangles all of same area, (2) same perimeter (3) same base angle, ... there seem to be many possibilities, at least some of which might be interesting.

The trigger for the present post, a lot of which is repeat: It was recently noted that The 'fair partition' question has a variant: For any n, can a given convex region be cut into n convex pieces all of same perimeter and diameter? The techniques used to answer the basic fair partition question for all n do not work readily for this variant. As far as I know, the perimeter-diameter and similar variants are now being investigated by experts.
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Another not very closely related query - on cutting convex n-gons into equal diameter triangles - was recently recorded here: https://mathoverflow.net/questions/392596/cutting-convex-polygons-into-triangles-of-same-diameter. An earlier post here on a related matter is: https://nandacumar.blogspot.com/2020/11/some-thoughts-on-monskys-theorem.html
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A bit added on June 6th 2021:

One more question:
Can any triangle (similarly, any quadrilateral) be cut into any finite number of mutually non-congruent triangles all of same area and perimeter?

Partial Answer: See https://arxiv.org/pdf/1711.04504.pdf
Theorem 6 there proves: "Let k ≥ 4. In any tiling of a convex k-gon with finitely many triangles, there are two triangles that share an edge." And if two triangles of same area and perimeter also share an edge, they are congruent. So, the answers for quads upwards is known and negative. For triangles, things are open, as of now!