Representations are sheaves conjecture for Legendrian links

Let $n\geq 1$, let $\mathbbm{k}$ be a field, and let $\Lambda\subset\mathbb{R}^3$ be a Legendrian link. The representation–sheaf conjecture. The $A_\infty$ category $\mathcal{R}ep_n(\Lambda,\mathbbm{k})$ of $n$-dimensional representations of the Chekanov–Eliashberg differential graded algebra of $\Lambda$ is $A_\infty$ equivalent to the category $\mathcal{S}h_n(\Lambda,\mathbbm{k})$ of constructible sheaves with microsupport on $\Lambda$ and microlocal rank $n$:

$$\mathcal{R}ep_n(\Lambda,\mathbbm{k})\stackrel{\sim}{\to}\mathcal{S}h_n(\Lambda,\mathbbm{k}).$$

This generalizes the known correspondence between the positive augmentation category and the sheaf category of microlocal rank one. The conjecture asks for the categorical correspondence for arbitrary representation dimension $n$ and remains unresolved in the supplied source.