Consider the discrete magnetic Laplacian on $\mathbb Z^2.$

$$(\Delta_{\alpha,\lambda}\psi)(n_1,n_2) = e^{-i \pi \alpha n_2} \psi(n_1+1,n_2) + e^{i\pi \alpha n_2} \psi(n_1-1,n_2) + \lambda \left(e^{i \pi \alpha n_1} \psi(n_1,n_2+1)+e^{-i\pi \alpha n_1} \psi(n_1,n_2-1) \right)$$

We consider $\alpha$ irrational. It is known (to me) that the spectrum of this operator as a set always coincides with the spectrum of the Almost Mathieu operator and that this operator has no point spectrum. But I am curious whether it is known when the spectrum (depending on $\lambda$) of this operator is absolutely continuous/singular continuous. Apparently, it is singular continuous if $\lambda =1$ as the measure of the spectrum is zero in this case.