Strong formality conjecture for the unit-generated subcategories

Let $X_q$ be the half-quantum flag variety, let $\widetilde{\mathcal{N}}$ be the Springer resolution, let $\mathcal{N}$ be the nilpotent cone, and use $\operatorname{QCoh}_{\mathrm{dg}}$, $\operatorname{IndCoh}_{\mathrm{dg}}$, and $\operatorname{Rep}_{\mathrm{dg}}$ for the derived categories described in the source. Strong formality conjecture. There is a central, fully faithful, $\operatorname{QCoh}_{\mathrm{dg}}(\check{G}/\check{B})$-linear tensor functor

$$\eta^*:\operatorname{QCoh}_{\mathrm{dg}}(\widetilde{\mathcal{N}})\to\operatorname{IndCoh}_{\mathrm{dg}}(X_q),$$

which is an equivalence onto the localizing $\operatorname{QCoh}_{\mathrm{dg}}(\check{G}/\check{B})$-submodule generated by the unit. There is likewise a fully faithful tensor functor

$$\bar{\eta}^*:\operatorname{QCoh}_{\mathrm{dg}}(\mathcal{N})\to\operatorname{Rep}_{\mathrm{dg}}(u(G_q)),$$

which is an equivalence onto the localizing subcategory generated by the unit. This is the detailed unit-generated formulation of the strong formality conjecture and implies the corresponding monoidal relationship between the Springer resolution and the small quantum group.