Transverse intersection conjecture for dimension functions of quadratic Sturmian spectra

For each quadratic irrationality $\alpha$, let $\Sigma_{\alpha,\lambda}$ be the spectrum of the corresponding Sturmian Hamiltonian at coupling constant $\lambda$. Transverse intersection conjecture. There exist quadratic irrationalities $\alpha_1$ and $\alpha_2$ such that the graphs of

$$\lambda\mapsto\dim_H\Sigma_{\alpha_1,\lambda}$$

and

$$\lambda\mapsto\dim_H\Sigma_{\alpha_2,\lambda}$$

intersect transversely at some point $\lambda_0$. Such an intersection would provide the crossing mechanism needed to produce non-differentiability in an associated non-stationary dimension function; analytic dependence is known for each individual quadratic irrationality.