Isogeny classification conjecture for Denjoy systems

Let $(\mathcal{D}_{F_k},F_k)$, for $k\in\{0,1\}$, be Denjoy systems with invariants $(\rho_k,Q_k)$. For $A\subset\mathbb{T}$, write $\mathbb{Q}A\subset\mathbb{T}$ for the $\mathbb{Q}$-submodule generated by $A$, and let $\equiv$ denote equality of subsets of $\mathbb{T}$ up to rotations. Denjoy isogeny classification conjecture. The two Denjoy systems are isogenous if and only if there exists $M=(m_{ij})\in\operatorname{PGL}_2(\mathbb{Q})$ such that

$$\rho_1=\frac{m_{11}\rho_0+m_{12}}{m_{21}\rho_0+m_{22}}$$

and

$$\mathbb{Q}Q_1\equiv\frac{1}{m_{21}\rho_0+m_{22}}\mathbb{Q}Q_0.$$

This is proposed as a classification of Denjoy systems up to isogeny, generalizing the proved Sturmian classification; no resolution is given for the general Denjoy case.