Augmentations are sheaves for Legendrian graphs

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical isotopy invariants: augmentation category, unital $A_{\infty}$-category, which lifts set of augmentations Chekanov-Eliashberg DGA, DG category constructible sheaves on front plane, with micro-support at contact infinity controlled by graph. other words, generalizing [21], prove are in singular case.