Monoidal recovery conjecture for the principal block

Let $\widetilde{\mathcal{N}}$ be the Springer resolution, let $u(G_q)$ be the small quantum group, let $\operatorname{PrinBlock}_{\mathrm{dg}}(u(G_q))$ be its principal block, and suppose the monoidal correspondence has functors $\eta^*$ from the Springer resolution to $X_q$ and $\kappa^*$ from the small quantum group representations to $X_q$. Monoidal recovery conjecture. The push-pull functor restricts to an equivalence

$$\eta_*\kappa^*:\operatorname{PrinBlock}_{\mathrm{dg}}(u(G_q))\overset{\sim}{\longrightarrow}\operatorname{QCoh}_{\mathrm{dg}}(\widetilde{\mathcal{N}}).$$

This is proposed to recover the equivalence of Arkhipov–Bezrukavnikov–Ginzburg and Bezrukavnikov–Lachowska in a manifestly monoidal way.