Non-differentiability conjecture for dimensions of bounded-type Sturmian spectra

Let $\Sigma_{\alpha,\lambda}$ be the spectrum of the Sturmian Hamiltonian with frequency $\alpha$ and coupling constant $\lambda$. Bounded-type non-differentiability conjecture. There exists a frequency $\alpha$ of bounded type such that

$$\lambda\mapsto\dim_H\Sigma_{\alpha,\lambda}$$

is not differentiable at at least one point $\lambda_0$. This is proposed as a consequence of extending the paper's non-stationary hyperbolic argument beyond the quadratic irrational case; analytic dependence is known for quadratic irrational frequencies, while the corresponding almost-sure regularity is described as conjectural.