The three-dimensional second-gap Mourre conjecture

Fix $\kappa=4$ and let $\{\mathcal{E}_n\}$ be the sequence from the stated theorem. For each $n\geq1$, define

$$A_{j_q\kappa}=\sum_{i=1}^{3}A_i(j_q,\kappa).$$

Three-dimensional Mourre conjecture. On every interval $(\mathcal{E}_n,\mathcal{E}_{n-1})$, the Mourre estimate holds with

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

where the coefficients are exactly those used in the two-dimensional case. In particular, $\{\mathcal{E}_n\}_{n=1}^{\infty}=J_2\cap\boldsymbol{\Theta}_{\kappa}(D)$. The claim is based on graphical evidence and is explicitly described as mysterious; no proof is supplied.