It is so annoying that I cannot pin this done. Let me try again.

Let's say, just in the hypertoric case (which I know I am cheating, but whatever). You have two spaces, why there is a matching of Fukaya categories for two seemingly unrelated spaces? Even just for $T^*P^{n-1}$ and $\C^2 / \mu_n$ resolution, why?

I know you will say, Gale duality, but I don't understand why it worked.

Let's try to just guess, what the Lagrangian correspondence will look like.

We have complex moment maps, $\mu: M = T^*\P^2 \to \C^2$ and $\mu^\vee: M^\vee = \C^2/\mu_3 \to \C$.