Unique hypotenuse multiplicity conjecture for congruent number triangles

Let $t$ be a congruent number. A primitive right integer triangle has squarefree part of its area equal to $t$ when its area is $t u^2$ for some integer $u$, and two such triangles are dissimilar when they are not similar. A hypotenuse multiplicity counts dissimilar primitive right integer triangles sharing the same hypotenuse.

Unique hypotenuse multiplicity conjecture. There do not exist two dissimilar primitive right triangles with the same hypotenuse and whose squarefree parts of the areas are equal to each other.

This conjecture asserts that every hypotenuse in the set associated with a congruent number occurs with multiplicity exactly one. The paper reports numerical experimentation supporting the conjecture; its general validity is not established.