Some thoughts on Monsky's Theorem

Monsky's Theorem: It is not possible to dissect a square into an odd number of triangles all of equal area.

Definitions: The diameter of a planar convex region is the greatest distance between any pair of points in the region. The least width of a 2D convex region can be defined as the least distance between any pair of parallel lines that touch the region.

Questions: Is it possible to dissect a square into any odd number of triangles all of equal perimeter/ equal diameter/ equal least width? Do negative answers to these questions follow from any proof(s) of Monsky's theorem?

Found this earlier discussion that addresses the equal perimeter question: https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter. There, Christian Blatter says: "The proof (of Monsky's theorem) is not easy, and some authors even take recourse to the axiom of choice for a proof. Anyway: To obtain a theorem about an odd number of triangles of equal perimeter should be even more difficult since square roots are entering the picture."

It is also seen quite easily that a square can be divided into n equal diameter triangles very easily for any n (for even n, divide the square into n/2 rectangular strips and cut each strip with a diagonal in half; for odd n, divide the square into n-2 isosceles triangles and 2 right triangles all with same diameter).

That leaves only the partition of the square into n triangles all of same least width.

Note added on June 6th 2021:

One more question: Is there any quadrilataral at all that allows partition into n equal area triangles where n is *any* integer? Or is the square/rectangle that quad that gives an upper bound on the possible n values? And which quad gives the lower bound?