Gorsky–Negut–Rasmussen conjecture for sheaves on Hilbert schemes

Let $\beta$ be a braid on $n$ strands. Let $\operatorname{Hilb}^n(\mathbb{C}^2,\mathbb{C})$ be the indicated Hilbert scheme, let $\mathcal{T}$ be its tautological rank-$n$ bundle, and let $\mathcal{F}_{\beta}$ be a $\mathbb{C}^*\times\mathbb{C}^*$-equivariant coherent sheaf. Gorsky–Negut–Rasmussen conjecture. The sheaf $\mathcal{F}_{\beta}$ has properties (a)–(e):

(a) $\operatorname{HHH}(\beta)\simeq H^*_{\mathbb{C}^*\times\mathbb{C}^*}\left(\operatorname{Hilb}^n(\mathbb{C}^2,\mathbb{C}),\mathcal{F}_{\beta}\otimes\wedge^{\bullet}\mathcal{T}^{\vee}\right)$ as triply graded vector spaces, with the stated grading correspondence;

(b) the action of symmetric functions in $x_i$ corresponds to the action of $\mathbb{C}[x_1,\ldots,x_n]^{S_n}$ on the right;

(c) for a braid whose closure is a knot, all $x_i$ act identically and $\mathcal{F}_{\beta}$ is supported on $\operatorname{Hilb}^n(\mathbb{C}^2,0)\times\mathbb{C}$;

(d) adding $\mathrm{FT}=(\sigma_1\cdots\sigma_{n-1})^n$ corresponds to tensoring $\mathcal{F}_{\beta}$ by $\mathcal{O}(1)$;

(e) $\mathcal{F}_{\beta}$ extends to a sheaf on the whole $\operatorname{Hilb}^n(\mathbb{C}^2)$ corresponding to $y$-ified homology. The conjecture was mostly proved in a series of papers by Oblomkov and Rozansky, so its database status is solved.