Koszul duality patterns in Floer theory

We study symplectic invariants of the open symplectic manifolds $X_{\mathrm{\Gamma }}$ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree $\mathrm{\Gamma }$. For any tree $\mathrm{\Gamma }$, we calculate (DG-)algebra models of the Fukaya category $\mathcal{F}(X_{\mathrm{\Gamma }})$ of closed exact Lagrangians in $X_{\mathrm{\Gamma }}$ and the wrapped Fukaya category $\mathcal{W}(X_{\mathrm{\Gamma }})$. When $\mathrm{\Gamma }$ is a Dynkin tree of type $A_n$ or $D_n$ (and conjecturally also for $E_6,E_7,E_8$), we prove that these models for the Fukaya category $\mathcal{F}(X_{\mathrm{\Gamma }})$ and $\mathcal{W}(X_{\mathrm{\Gamma }})$ are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of $X_{\mathrm{\Gamma }}$ for $\mathrm{\Gamma }=A_n,D_n$, based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.