Flattening conjecture for Gorsky–Negut–Rasmussen's tautological complex

Let ${\mathcal{V}}$ be the Gaitsgory central complex for the standard representation, let $\mathcal{E}$ be a complex, and let the requirements of Gorsky–Negut–Rasmussen's conjecture refer to the filtration and other properties required of the image of the tautological bundle on the flag Hilbert scheme.

Flattening conjecture. The flattening of ${\mathcal{V}}$ is a complex ${\mathcal{E}}$ satisfying the requirements of Gorsky–Negut–Rasmussen's conjecture.

The claim gives a proposed construction of the complex whose existence is required in the Gorsky–Negut–Rasmussen framework. It is easy to verify for $n=2$, while the supplied passage does not state a resolution in general.