Screening-operator identification conjecture for Fock representations

Let $\mathcal{F}_{\Lambda}$ be the Fock representation with screening operators $\xi_i$ and $\eta_i$ for $i\in\hat{I}^-\cup\{m\}$, and set

$$\xi=\prod_{i\in\hat{I}^-\cup\{m\}}\xi_i,\qquad \eta=\prod_{i\in\hat{I}^-\cup\{m\}}\eta_i.$$

For a weight $\Lambda$, let $V(\Lambda)$ denote the irreducible highest weight $U^{ver}_q\widehat{\mathfrak{gl}}_{m|n}$-module with highest weight $\Lambda$. Screening-operator identification conjecture. The following identifications hold:

$$V(\Lambda_i)=\ker\eta=\eta\xi\mathcal{F}_{\Lambda_i}\quad (i\in I),$$ $$V((1-a)\Lambda_0+a\Lambda_m)=\mathcal{F}_{(1-a)\Lambda_0+a\Lambda_m}\quad (a\in\mathbb{C}\setminus\mathbb{Z}),$$ $$V((1-a)\Lambda_0+a\Lambda_m)=\operatorname{coker}\eta=\xi\eta\mathcal{F}_{(1-a)\Lambda_0+a\Lambda_m}\quad (a\in\mathbb{Z}_{>0}),$$ $$V((1-a)\Lambda_0+a\Lambda_m)=\ker\eta=\eta\xi\mathcal{F}_{(1-a)\Lambda_0+a\Lambda_m}\quad (a\in\mathbb{Z}_{\leq 0}).$$

These identifications describe the irreducible highest weight modules as kernels, cokernels, or full Fock representations according to the parameter. The source attributes this conjectural identification to prior work and does not state a resolution.