Strong formality conjecture for the half-quantum flag variety

Let $G$ be an almost-simple algebraic group with dual group $\check{G}$, let $q$ be a chosen complex root of unity, let $X_q$ be the half-quantum flag variety, let $\widetilde{\mathcal{N}}$ be the Springer resolution, and let $\mathcal{N}$ be the nilpotent cone. Write $\operatorname{QCoh}_{\mathrm{dg}}(Y)$ for the derived infinity-category of quasi-coherent sheaves on $Y$, $\operatorname{IndCoh}_{\mathrm{dg}}(X_q)$ for the formal cocompletion of coherent dg sheaves, and $\operatorname{Rep}_{\mathrm{dg}}(u(G_q))$ for the formal cocompletion of finite-dimensional dg representations of the small quantum group. Strong formality conjecture. There is a $\operatorname{QCoh}_{\mathrm{dg}}(\check{G}/\check{B})$-linear, fully faithful, central tensor functor

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

and a corresponding fully faithful tensor functor

$$\operatorname{QCoh}_{\mathrm{dg}}(\mathcal{N})\to\operatorname{Rep}_{\mathrm{dg}}(u(G_q)).$$

The conjecture is motivated by the enhanced derived category of $X_q$, the computation relating its derived endomorphisms to the Springer resolution, and known formality results.