Conformal-block correspondence under quantum Drinfeld–Sokolov reduction

Let ${\mc C}$ be a smooth projective algebraic curve, let $p_i$ be marked points, let $L_{\lambda_i,{}^Lk}$ be the relevant modules for the affine algebra at level ${}^Lk$, and let $H_{\operatorname{DS}}(L_{\lambda_i,{}^Lk})$ be their quantum Drinfeld–Sokolov reductions. Let C_{V_{{}^Lk({}^L\mathfrak{g})}({\mc C},(p_i),(L_{\lambda_i,{}^Lk})) and $C_{{\mc W}_\beta(\mathfrak{g})}({\mc C},(p_i),(H_{\operatorname{DS}}(L_{\lambda_i,{}^Lk})))$ denote the corresponding spaces of conformal blocks. Conformal-block correspondence conjecture. Provided that the parameters satisfy the conditions of the braided tensor equivalence conjecture, there are isomorphisms

$$C_{V_{{}^Lk({}^L\mathfrak{g})}}({\mc C},(p_i),(L_{\lambda_i,{}^Lk}))\simeq C_{{\mc W}_\beta(\mathfrak{g})}({\mc C},(p_i),(H_{\operatorname{DS}}(L_{\lambda_i,{}^Lk}))).$$

This is described as a global consequence of the local categorical statement, explicitly at least for ${\mc C}=\mathbb{C}\mathbb{P}^1$ in the surrounding discussion. It remains conditional on the braided tensor equivalence conjecture.