Bellissard's almost-everywhere constancy conjecture for Sturmian spectrum dimension

Let $H_{\lambda,\alpha,\omega}$ be the Sturmian Hamiltonian with coupling $\lambda>0$, irrational frequency $\alpha\in(0,1)$, and phase $\omega\in S^1$, and let $\sigma_{\lambda,\alpha}$ denote its spectrum. Bellissard's conjecture. For each $\lambda>0$, the Hausdorff dimension of the spectrum of $H_{\lambda,\alpha,\omega}$ is $\alpha$-constant Lebesgue almost everywhere. This conjecture concerns the dimensional regularity of Sturmian Schrödinger spectra; the supplied text does not state whether it has been resolved.