Gauss-map diffeomorphism conjecture for Sturmian spectra

Let $d706>0$ and $d6fc\in(0,1)\setminus\mathbb{Q}$, and let $G(\alpha)$ denote the Gauss map applied to $d6fc$. Write $d6a9_{\lambda,\alpha}$ and $d6a9_{\lambda,G(\alpha)}$ for the corresponding Sturmian spectra. Gauss-map diffeomorphism conjecture. For every $d706>0$, the spectra $d6a9_{\lambda,\alpha}$ and $d6a9_{\lambda,G(\alpha)}$ are diffeomorphic: there exist neighborhoods $U\supsetd6a9_{\lambda,\alpha}$ and $V\supsetd6a9_{\lambda,G(\alpha)}$ and a $C^1$ diffeomorphism $f:U\to V$ such that $f(d6a9_{\lambda,\alpha})=d6a9_{\lambda,G(\alpha)}$. This is presented as a further natural conjecture related to the Gauss-map invariance underlying the dimension result.