Quasi-integrability of the multidimensional numen in the noncommutative case

Let $H$ be as in the source's theorem giving the multidimensional $\mathcal{F}$-series for $\chi_H$, and suppose that $\mathbf I_d-\alpha_H(\mathbf 0)$ is invertible. Let $\psi_H$ be the auxiliary function and let $L_H$ be the operator appearing in the source's displayed formula. Noncommutative quasi-integrability conjecture. The displayed formula for $L_H$ applied to the Fourier transform of the auxiliary function gives a map

$$\widehat{\mathbb{Z}}_p^{r}\to\overline{\mathbb{Q}}^{d,d}$$

which is a Fourier transform of $\psi_H$ with respect to the standard $(p,q_H)$-adic frame. This would imply quasi-integrability of $\chi_H$; the source states that no rigorous argument is currently available.