Gorsky–Neguț–Rasmussen conjecture on coherent computation of Khovanov–Rozansky homology

Let $F_\beta$ be the Rouquier complex associated to a braid $\beta$, let $\mathcal{T}_n$ be the tautological bundle on the dg flag Hilbert scheme, and let $\iota_*$ and $\iota^*$ be the adjoint functors in the preceding conjecture. Write $\int$ for pushforward to a point in the coherent equivariant category.

Gorsky–Neguț–Rasmussen conjecture. There is an isomorphism of bigraded vector spaces

$$\operatorname{HHH}^{k}(F_\beta)\simeq\operatorname{HH}^{0}\bigl(\iota^*(\wedge^k\mathcal{T}_n^{\vee})\otimes F_\beta\bigr)\simeq\int \iota_*(F_\beta)\otimes\wedge^k\mathcal{T}_n^{\vee}.$$

The proposed formula would compute Khovanov–Rozansky homology on the coherent side of the flag-Hilbert-scheme model. The paper describes it as a prediction from the cited conjecture, not as a theorem.