Koszul duality via suspending Lefschetz fibrations

Let $M$ be a Liouville 6-manifold which is the smooth fiber of a Lefschetz fibration on ${\mathbb{C}}^4$ constructed by suspending a Lefschetz fibration on ${\mathbb{C}}^3$. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category $\mathcal{F}(M)$ and the wrapped Fukaya category $\mathcal{W}(M)$ are related through $A_{\mathrm{\infty }}$-Koszul duality, by identifying them with cyclic and Calabi-Yau completions of the same quiver algebra. This implies the split-generation of the compact Fukaya category $\mathcal{F}(M)$ by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi-dilations in the sense of Seidel-Solomon are obtained.