Simon's phase-insensitivity conjecture for singular spectra

Let $H_{\alpha,\theta}$ be the quasi-periodic Schrödinger operator in the paper, with phase $\theta\in\mathbb{T}^{\nu}$, and write

$$\sigma_{sing}(H_{\alpha,\theta})=\sigma_{sc}(H_{\alpha,\theta})\cup\sigma_{pp}(H_{\alpha,\theta}).$$

Simon's conjecture. For all $d,v\in\mathbb{N}$ and all $\theta\in\mathbb{T}^{\nu}$, $\sigma_{sing}(H_{\alpha,\theta})$ is constant in $\theta$.

The absolutely continuous spectrum is known to be phase-insensitive in the stated setting, whereas the singular continuous and pure point components can vary with the phase; the conjecture asserts that their union nevertheless does not. Its status is not resolved in the supplied source.