Conjecture on uniqueness of root-of-unity conformal-block solutions

Let $q=\exp(\pi i/\ell)$ with $\ell\ge3$, and let $\operatorname{Conf}_{\vec\Lambda}^{(\ell)}$ be the space spanned by the lattice restrictions of the Weyl anti-symmetrized hypergeometric conformal blocks. The modified Macdonald–Ruijsenaars operator $\widetilde{\mathbb M}_\Theta$ acts on functions $\mathbb Z\to L_{\vec\Lambda}[0]$, and the relevant quasi-periodicity is $\ell$-periodicity up to the sign $(-1)^{{\mathfrak m}+1}$. Root-of-unity conformal-block uniqueness conjecture. If $\tilde\vartheta:\mathbb Z\to L_{\vec\Lambda}[0]$ satisfies

$$\widetilde{\mathbb M}_\Theta\tilde\vartheta(\delta)=(\dim_q L_\Theta)\tilde\vartheta(\delta)$$

and

$$\tilde\vartheta(\delta+\ell)=(-1)^{{\mathfrak m}+1}\tilde\vartheta(\delta),$$

then $\tilde\vartheta(\delta)\in\operatorname{Conf}_{\vec\Lambda}^{(\ell)}$. The conjecture is the root-of-unity analogue of the generic uniqueness statement; the modified operator and quasi-periodicity make the assertion meaningful on the lattice, but the general result remains open.