  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 $A_\infty$-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.
