Finiteness conjecture for rational triangles fixed up to similarity by a weakly metric affine transformation

Let $V$ be the set of weakly metric affine transformations, let $f_I$ denote the identity transformation, and let $X(T)$ denote the set associated with a rational triangle $T$. Two rational triangles are dissimilar when they are not similar, and write $f(T)\sim T$ when $f(T)$ and $T$ are weakly metric equivalent.

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 $f(T)\sim T$.

This conjecture addresses the question of whether a nonidentity weakly metric affine transformation can preserve the weak metric equivalence class of only finitely many rational triangles up to similarity. The source provides no resolution, so the conjecture remains open.