Kotani–Last conjecture for almost periodic Dirac operators

Let $\Lambda_\varphi$ be the one-dimensional Dirac operator on $L^2({\mathbb{R}},{\mathbb{C}}^2)$, with potential $\varphi:{\mathbb{R}}\to{\mathbb{C}}$ satisfying

$$\sup_{x\in{\mathbb{R}}}\int_x^{x+1}|\varphi(t)|^2\,dt<\infty.$$

Call $\Lambda_\varphi$ almost periodic when its potential is uniformly almost periodic, meaning that for every $\epsilon>0$ the set

$$\{\tau\in{\mathbb{R}}:\|\varphi(x+\tau)-\varphi(x)\|_\infty<\epsilon\}$$

is relatively dense in ${\mathbb{R}}$. Kotani–Last conjecture. The presence of absolutely continuous spectrum implies almost periodicity of the potential. This conjecture is refuted: counterexamples are known for continuum and discrete Schrödinger operators, reflectionless Jacobi matrices, and, as discussed in the source, the Dirac setting.