Dry Ten Martini conjecture for type-I operators

Let $\alpha$ be irrational and let $v\in C^\omega(\mathbb{T},\mathbb{R})$. Write $\Sigma_{v,\alpha}$ for the spectrum and $N_{v,\alpha}$ for the integrated density of states. An energy $E_k\in\Sigma_{v,\alpha}$ is type-I when its T-acceleration satisfies $\overline{\omega}(E_k)=1$. Dry Ten Martini conjecture. For any irrational $\alpha$ and any $v\in C^\omega(\mathbb{T},\mathbb{R})$, each type-I energy $E_k\in\Sigma_{v,\alpha}$ satisfying

$$N_{v,\alpha}(E_k)\equiv k\alpha\pmod{\mathbb{Z}}$$

is a boundary of an open gap. This is a type-I form of the Dry Ten Martini Problem, asserting that every gap allowed by the gap-labeling condition is open in this regime. A complete characterization of the spectral gaps remains open.