Conjectured conjugate operators for Mourre estimates on discrete Schrödinger bands

Fix $\kappa\geq 2$, and let $\{\mathcal{E}_n\}$ be the sequence from Theorem 1.7 of the cited source. For each interval $(\mathcal{E}_n,\mathcal{E}_{n-1})$ with $n\geq 1$, define

$$\mathbb{A}(n)=\sum_{q=1}^{N(n)}\rho_{j_q\kappa}(n)A_{j_q\kappa},\qquad A_{j_q\kappa}=\sum_{1\leq i\leq 2}A_i(j_q,\kappa).$$

Conjugate-operator conjecture. There exists such a conjugate operator $\mathbb{A}(n)$ for which the Mourre estimate for $\Delta$ holds with respect to $\mathbb{A}(n)$ for every $E\in(\mathcal{E}_n,\mathcal{E}_{n-1})$. The operator $\mathbb{A}(n)$ is typically not unique, can be chosen with $N(n)=2n$, and consequently

$$\{\mathcal{E}_n\}=J_2\cap\boldsymbol{\Theta}_{\kappa}(\Delta),\qquad \forall\kappa\geq 2.$$

This conjecture concerns the existence of conjugate operators yielding strict Mourre estimates on the spectral bands between consecutive thresholds. The source presents numerical and graphical evidence but does not establish the conjecture in general.