Irreducibility conjecture for imaginary Whittaker modules

Let $\bmod\text{-}\mathcal{H}_q^{l}$ be the subcategory of $\mathcal{H}_q$-modules on which $\gamma=q^l\operatorname{Id}$, and let $\mathbb{I}_q^\lambda(V)=\mathcal{U}_q(\widehat{\mathfrak{g}})\otimes_{B_q^d}V$. For $(\zeta,a)\in(\operatorname{Hom}_{\mathrm{alg}}(\mathcal{H}_q^\prime(+),\mathbb{C}(q^{1/2})),\mathbb{C}^{\times})$ and $\lambda\in\dot{P}$, write $\widehat{M}_{\zeta,\lambda}^q=\mathbb{I}_q^\lambda(M_{\zeta,a}^q)$ and $\widehat{K}_{\zeta,\lambda,t}^q=\mathbb{I}_q^\lambda(K_{\zeta,a,t}^q)$. Irreducibility conjecture. If $l=a$, then the $\mathcal{U}_q(\widehat{\mathfrak{g}})$-modules $\widehat{M}_{\zeta,\lambda}^q$ and $\widehat{K}_{\zeta,\lambda,t}^q$ are irreducible. The surrounding section studies irreducibility of imaginary Whittaker modules over quantum affine algebras; the parser supplies no evidence resolving this assertion.