Frenkel–Gaitsgory conjecture for graded opers

Let $\slam\in \bar P^+$, and let $\KLlam$ be the full subcategory of

$${\mathcal O}^{\operatorname{KL}}_{\operatorname{crit}}$$

consisting of modules annihilated by $\ker\chilam$. For $\lambda\in \Paf{\operatorname{crit}}$, let $V(\lambda)$ be the corresponding Verma module and $L(\lambda)$ its simple quotient.

Frenkel–Gaitsgory conjecture. The category $\KLlam$ is semisimple for every $\slam\in \bar P^+$, and, for each $\lambda\in \Paf{\operatorname{crit}}$, there is an isomorphism

$$V(\lambda)/\ker \chilam\cdot V(\lambda)\cong L(\lambda).$$

The conjecture concerns the structure of critical-level representations with a fixed graded central character; it was announced by Frenkel and Gaitsgory. Its resolution status is not specified in the supplied text.