Seidel's long exact sequence conjecture for composed Dehn twists

Let $(M,\omega)$ be a symplectic manifold of dimension $2d$ with $2c_1(M,\omega)=0$, and let $L_1,\dots,L_n$ be graded Lagrangian spheres. Let $\mathcal{A}$ be the directed Fukaya category with morphism complexes

$$\operatorname{hom}_{\mathcal{A}}(L_j,L_k)=\begin{cases}CF(L_j,L_k)&j<k,\\ \mathbb{K}e_j&j=k,\\ 0&j>k.\end{cases}$$

Set $\tau=\tau_1\circ\cdots\circ\tau_n$. Seidel's conjecture. There is a long exact sequence

$$\cdots\rightarrow H^*(D)\rightarrow HF^*(\operatorname{id})\rightarrow HF^*(\tau)\rightarrow\cdots,$$

where $D$ is the cochain complex

$$\bigoplus_{1\leq k\leq n}\bigoplus_{1\leq i_1<\cdots<i_k\leq n}CF(L_{i_k},L_{i_1})\otimes CF(L_{i_{k-1}},L_{i_k})\cdots\otimes CF(L_{i_1},L_{i_2})[d+n]$$

with a differential analogous to the Hochschild differential. This conjecture seeks to compute the fixed-point Floer homology of the composed Dehn twists from the topology of $M$ and the directed Fukaya-category morphism complexes. The paper proves an exact sequence of this form, so the conjecture is resolved.