Eventual projective-integral equivalence conjecture for irrational parameters

Let $\alpha,\beta\in\mathbb{R}\setminus\mathbb{Q}$ be irrational. Say that they are eventually $\operatorname{PGL}_2(\mathbb{Z})$-equivalent if there exists $m\in\mathbb{N}$ such that, for every $n\geq m$, the numbers $n\alpha$ and $n\beta$ lie in the same $\operatorname{PGL}_2(\mathbb{Z})$ orbit. Eventual $\operatorname{PGL}_2(\mathbb{Z})$-equivalence conjecture. The numbers $\alpha$ and $\beta$ are eventually $\operatorname{PGL}_2(\mathbb{Z})$-equivalent if and only if

$$\alpha=\pm\beta\pmod{1}.$$

This is an arithmetic strengthening related to the classification of eventual flow equivalence for Sturmian systems; the source gives no resolution.