Hey, I made some progress today. About Koszul duality, at least, I have a concrete conjecture. Let's state it as 'The symplectic Koszul duality for category O'. It goes as following: take a 3d N=4 gauge theory, some compact group $G$ acting on some representation $T^*N$. You can form two LG A-models, let's call that $(M_{H, \alpha}, W_{H,\beta})$ and $M_{C, \beta}, W_{C, \alpha}$, where $\alpha$ and $\beta$ are some parameters. Then you prove that the two wrapped Fukaya categories are equivalent (after passing to $\Z/2$ grading? or invent a mixed version of the Fukaya category, where the hom space has a second grading)