3 problems

Kenyon–Smillie conjecture. Their computer search identified all triples of angles of acute and right-angled rational triangles that do not satisfy the criterion.

Finiteness conjecture. If $f\in V$ and $f\neq f_I$, there are only a finite number of dissimilar rational triangles $T\in X(T)$ such that $f(T)\sim T$.

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…