Small-coupling continuity conjecture for Sturmian spectrum dimension

For an irrational $\alpha\in(0,1)$, let $\sigma_{\lambda,\alpha}$ denote the corresponding Sturmian spectrum and let $\dim_H$ denote Hausdorff dimension. Small-coupling continuity conjecture. For any irrational $\alpha\in(0,1)$,

$$\lim_{\lambda\rightarrow 0}\dim_H\sigma_{\lambda,\alpha}=1.$$

The source states that this has been proved when $\alpha$ is an irrational of bounded type, while the case of general irrational $\alpha$ remains open.