Kac–Roan–Wakimoto–Arakawa conjecture on Drinfeld–Sokolov reduction

Let $\bar{\mathfrak{g}}$ be a finite-dimensional simple Lie algebra, let $f$ be a nilpotent element that is regular in a Levi subalgebra $\bar{\mathfrak{l}}$, and let $W_f$ be the Weyl group of that Levi subalgebra. Let $k$ be a boundary principal admissible level, let $L_k(\bar{\mathfrak{g}})$ be the corresponding simple affine vertex algebra, and let $\Psi_f^-:L_k(\bar{\mathfrak{g}})\operatorname{-mod}\nolimits\to \mathcal{W}_k(\bar{\mathfrak{g}},f)\operatorname{-mod}\nolimits$ be the negative quantized Drinfeld–Sokolov reduction functor. For $\lambda\in\operatorname{Adm}_k$, write $L(\lambda)$ for the corresponding simple module and let $\Phi^{\!\vee}_{\lambda,+}$, $\Phi^{\!\vee}_+$, and $\bar{\Phi}^{\!\vee}_f$ be the positive coroot sets appearing in the reduction criterion.

Kac–Roan–Wakimoto–Arakawa conjecture. The functor $\Psi_f^-$ is exact, and the image of every simple module is either simple or zero. Moreover,

$$\Psi_f^-(L(\lambda))\ne 0\quad\Longleftrightarrow\quad \Phi^{\!\vee}_{\lambda,+}\subset \Phi^{\!\vee}_+\backslash\bar{\Phi}^{\!\vee}_f.$$

Finally, for $\lambda,\lambda'\in\operatorname{Adm}_k$,

$$\Psi_f^-(L(\lambda))\simeq\Psi_f^-(L(\lambda'))\quad\Longleftrightarrow\quad \lambda=x.\lambda'\text{ for some }x\in W_f.$$

This predicts exactness and a complete description of the nonzero simple images and their identifications under negative Drinfeld–Sokolov reduction at boundary principal admissible levels. The statement is presented as expected in the source, and no resolution is supplied.