Self-duality conjecture for the Hilbert scheme under 3D-mirror symmetry

Let $\mathsf{X}=\operatorname{Hilb}^{n}(\mathbb{C}^{2})$, and let $\mathsf{X}^{!}$ denote its 3D-mirror dual. Let $\mathbf{U}(a,z)$ and $\mathbf{L}(a,z)$ be the matrices of twisted elliptic stable-envelope fixed-point components, let $\kappa^{*}$ be the parameter transformation

$$\kappa^{*}: a\mapsto z\sqrt{\hbar},\qquad \hbar\mapsto \frac{1}{\hbar},\qquad z\mapsto a\sqrt{\hbar},$$

and let $t$ denote matrix transposition. Self-duality conjecture. The Hilbert scheme is self-dual under 3D-mirror symmetry, $\mathsf{X}\cong\mathsf{X}^{!}$, and

$$\mathbf{U}(a,z)=\kappa^{*}\!\left(\mathbf{L}(a,z^{-1}\hbar^{-1})^{t}\right).$$

This is the fixed-point formulation of the paper’s mirror conjecture for the Hilbert scheme; its resolution is not specified in the supplied text.