Duality compatibility for Fukaya-category cobordism cone decompositions

Let $W$ be a Lagrangian cobordism with ends $L_1,\ldots,L_s$ and associated paths $\gamma_1,\ldots,\gamma_s$. Let $\mathcal{F}$ be the Fukaya category, let $\mathcal{M}_{W,\gamma_j}^l$ and $\mathcal{M}_{W,\gamma_j}^{r,\mathit{rel}}$ be the associated left and relative right $\mathcal{F}$-modules, and let $(\cdot)^\vee$ denote the dual module. Write $\delta_{W,j}$ for the quasi-isomorphisms induced by the relative weak Calabi--Yau pairing, and let $\nu_j$ and $\xi_j$ be the morphisms in the left-module and dual right-module exact triangles.

Duality compatibility conjecture. For $j=1,\ldots,s-1$, the quasi-isomorphisms

$$\delta_{W,j}:=(\mathbf{I}_{\gamma_j})^*(\delta_0^{\mathcal{F}_c})_W:\mathcal{M}_{W,\gamma_j}^l\to (\mathcal{M}_{W,\gamma_{j-1}}^{r,\mathit{rel}})^\vee$$

have the stated compatibility properties: $\delta_{W,1}=(\delta^\mathcal{F}_0)_{L_1}$; for $j=2,\ldots,s-1$, the diagrams comparing the left-module exact triangles with the duals of the right-module exact triangles commute in $\mathcal{F}\mathit{--}\operatorname{mod}$ up to homotopy of module morphisms; and the analogous final diagram for $L_s$ also commutes up to homotopy. Consequently, the induced diagrams commute in the derived category $D(\mathcal{F})$.

The conjecture predicts that the relative weak Calabi--Yau duality pairing is compatible with the cone decompositions associated to a Lagrangian cobordism. The paper states this expected comparison but does not carry out its details, so the compatibility remains open.