  We consider Takahashi's categorical interpretation of the Berglund-Hubsch
mirror symmetry conjecture for invertible polynomials in the case of chain
polynomials. Our strategy is based on a stronger claim that the relevant
categories satisfy a recursion of directed $A_{\infty}$-categories, which may
be of independent interest. We give a full proof of this claim on the B-side.
On the A-side we give a detailed sketch of an argument, which falls short of a
full proof because of certain missing foundational results in Fukaya-Seidel
categories, most notably a generation statement.
