Hilbert-scheme model for a wrapped unknot invariant

Let $\operatorname{Hilb}^n(\mathbb{C}^2)$ be the Hilbert scheme of $n$ points, let $\mathcal{I}$ be its tautological ideal sheaf, and let $X,Y$ be its two commuting endomorphisms. Assume a monoidal functor

$$\iota^*:D^b\operatorname{Coh}(\operatorname{Hilb}^n(\mathbb{C}^2))\to K^b(\operatorname{SBim}_n).$$

Let the $\mathfrak{gl}_N$ invariant of a single unknot wrapped around $n$ vertical strands be formed in the corresponding categorified braid setting. Hilbert-scheme invariant conjecture. This invariant is isomorphic to the $\mathfrak{gl}_N$ reduction of

$$\iota^*(\mathcal{I}/(Y,X^N)\mathcal{I}).$$

The statement would identify a concrete annular link invariant with an object obtained from tautological Hilbert-scheme data. It is one of the conjectural connections discussed with work of Gorsky, Negut, and Rasmussen; no resolution is given in the supplied text.