The Mourre-estimate conjecture for successive bands in the second gap

Fix $\kappa\geq4$ with $\kappa\in\mathbb{N}_e$, and let $\{\mathcal{E}_n\}$ be the sequence from the stated theorem. Let $A_{j\kappa}=\sum_{i=1}^{2}A_i(j,\kappa)$ be the conjugate operators used in the paper. Second-gap Mourre conjecture. For every $n\geq1$, there exists a conjugate operator

$$\mathbb{A}(n)=\sum_{q=1}^{N(n)}\rho_{j_q\kappa}(n)A_{j_q\kappa}$$

such that the Mourre estimate holds for every $E\in(\mathcal{E}_n,\mathcal{E}_{n-1})$. The operator is typically nonunique and can be chosen with $N(n)=2n$; in particular, $\{\mathcal{E}_n\}=J_2\cap\boldsymbol{\Theta}_{\kappa}(D)$. The claim is supported by numerical evidence, while the author explicitly reports no rigorous Mourre estimate on a new interval.