Speculations on homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$

Given an algebraic hypersurface $H=f^{-1}(0)$ in $({\mathbb{C}}^{\ast }{)}^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched version of this picture which also features the wrapped Fukaya category of the complement $({\mathbb{C}}^{\ast }{)}^n\setminus H$ and the Fukaya-Seidel category of the Landau-Ginzburg model $(({\mathbb{C}}^{\ast }{)}^n,f)$. We illustrate our speculations on simple examples, and sketch a proof of homological mirror symmetry for higher-dimensional pairs of pants.