At the risk of writing things that are obvious to those listening in: this is Nadler-land, no?

If $X$ is a smooth complex variety with reductive group $G$ acting, and $\mu_{\mathbb C}: T^*X\rightarrow {\mathfrak g}^*$ is the complex moment map, then $\mu_{\mathbb C}^{-1}(0)/G = T^*(X/G)$ provided one interprets all quotients as stacks.

If $T^*X$ is hyperkahler and we do the hyperkahler quotient for the maximal compact of $G$, picking a nontrivial real moment value $\mu_{\mathbb R}^{-1}(\zeta)$ at which to reduce amounts (by Kirwan) to imposing a GIT stability on $\mu_{\mathbb C}^{-1}(0)$---i.e. to picking a nice open subset of the cotangent stack $T^*(X/G)$ that is actually a variety. A stack version of Nadler's "microlocal branes" theorem would describe the (suitable, undoubtedly homotopical/derived) exact Fukaya category as the constructible derived category of $X/G$.

Since I'm completely ignorant of how the Nadler-Zaslow/Nadler story actually works, I'd like to then imagine that such an equivalence microlocalizes properly to give an equivalence over the hyperkahler reduction (i.e. the nice open set) as well? Admittedly, by microlocalizing to the stable locus one should avoid all the derived unpleasantness (this should be analogous to what happens in Bezrukavnikov-Braverman's proof of "generic" geometric Langlands for $GL_n$ in characteristic $p$, where by localizing to the generic locus, ${\mathcal D}$-module really means ${\mathcal D}$-module, not "module over the enveloping algebroid of the tangent complex" or something like that).

Admittedly, I don't have a clue how to deal with the issue that the base $X$ in the important examples is typically affine...maybe if one forces some kind of boundary conditions also in the $X$-direction one could make the Fukaya category nontrivial in Tim's example of the Hilbert scheme??