The fundamental-germ isomorphism conjecture for nonstandard integers

Let $r,s\in\mathbb R\setminus\mathbb Q$ be irrational numbers, and let ${}^{\ast}\mathbb Z_r$ and ${}^{\ast}\mathbb Z_s$ denote their associated groups of nonstandard diophantine approximations. Recall that $r$ and $s$ are equivalent if there exists $A\in{\sf SL}(2,\mathbb Z)$ such that $s=A(r)=(ar+b)/(cr+d)$. The fundamental-germ isomorphism conjecture. If

$${}^{\ast}\mathbb Z_r\cong{}^{\ast}\mathbb Z_s,$$

then $r$ and $s$ are equivalent. This would characterize the equivalence class of an irrational number through the isomorphism type of its fundamental germ, refining the known implication that equivalent irrational numbers have isomorphic groups; the statement is presented as an open conjecture.