$A_\infty$ Sabloff Duality via the LSFT Algebra

We use Ng's LSFT algebra to upgrade Sabloff duality of Legendrian knots to a quasi-isomorphism of $A_\infty$ bimodules over the positive augmentation category $\mathcal{A}ug_+$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of $\mathcal{A}ug_+$-bimodules, using a quotient category $\mathcal{C}$ of short Reeb chords. In addition, we define curved augmentations of the LSFT algebra and show that they can be used to construct a homotopy inverse of the $A_\infty$ Sabloff map, together with all higher homotopies. The above results suggest a conjectural recipe for an explicit weak relative Calabi-Yau structure on the quotient $A_\infty$ functor $\pi:\mathcal{A}ug_+\to \mathcal{C}$.