Conjecture on the minus Drinfeld–Sokolov reduction of category O modules

Let $\bar{\ta{ g}}$ be the finite-dimensional simple Lie algebra underlying the affine algebra, let $\bar{\rho}$ be its Weyl vector, let $\bar{\ta{ h}}^*$ be the corresponding finite-dimensional weight space, and let $\bar{\ta{ \Delta}}_+$ be the set of positive finite roots. For a level $\tau\ta{ \kappa}$, write $\ta{ O}_{\tau\ta{ \kappa}}$ for the relevant category of modules, let $H^i_-$ denote minus Drinfeld–Sokolov reduction, and let $L(\ta{ \lambda})$ and ${\bf L}(\gamma_{\bar{\ta{ \lambda}}})$ denote the corresponding simple modules. A weight $\ta{ \lambda}$ is non-critical when it does not lie on the critical hyperplane.

Minus-reduction conjecture. For any $\ta{ \kappa}\in\mathbb C$ and any V\in\operatorname{Obj}\ta{ O}_{\ta{ \kappa}},

$$H^i_-(V)=0\qquad (i\ne 0).$$

For any non-critical weight $\ta{ \lambda}$,

$$H^0_-(L(\ta{ \lambda}))\cong \begin{cases} {\bf L}(\gamma_{\bar{\ta{ \lambda}}})&\text{if }\langle \ta{ \lambda}+\rho,\bar{\ta{ \alpha}}^{\vee}\rangle\notin\{1,2,\dots\}\text{ for all }\bar{\ta{ \alpha}}\in\bar{\ta{ \Delta}}_+,\\ 0&\text{otherwise.} \end{cases}$$

This predicts exactness of minus Drinfeld–Sokolov reduction on the specified category and identifies its effect on simple modules. The supplied text gives no resolution evidence, so the conjecture is recorded as open.