Quantized conductance conjecture for the almost Mathieu operator

Let $H_0$ be the Hamiltonian whose potential on the sites $n$ is the quasi-periodic function $v(n)=U\cos(2\pi\omega n)$, where $-2<U<2$ and $\omega$ is irrational. Assume that the Fermi level lies inside the absolutely continuous spectrum of $H_0$ in the infinite-length limit. Quantized conductance conjecture. The conductance $g_j$ in the infinite-length limit is quantized as

$$g_j=\frac{1}{2\pi}.$$

This conjecture concerns the quantization of conductance for the Peierls–Harper, or almost Mathieu, model. The preceding periodic case establishes the analogous quantization under stronger spectral assumptions, while the quasi-periodic absolutely continuous-spectrum setting remains the conjectural extension.