Sheaf quantization and Fukaya category conjectures for Liouville manifolds

Let $X$ be a Liouville manifold. For a Lagrangian brane $\bm L$ in $X$, let $CF_{\mathrm{SQ}}(\bm{L}, \bm{L},\sigma)$ denote the curved $A_\infty$-algebra of sheaf quantizations and let $CF(\bm{L}, \bm{L},\sigma)$ denote the corresponding Fukaya-theoretic algebra. Let $\Fuk(X)$ be the infinitesimally wrapped Fukaya category over $\Lambda_0$, and let $\mu(X,u<\infty)$ be the microlocal category.

Sheaf quantization and Fukaya category conjectures. For each Lagrangian brane $\bm L$ in $X$, there is an almost quasi-isomorphism

$$CF_{\mathrm{SQ}}(\bm{L}, \bm{L},\sigma)\simeq CF(\bm{L}, \bm{L},\sigma).$$

Moreover, there exists an almost fully faithful embedding

$$F\colon \Fuk(X)\hookrightarrow \mu(X, u<\infty)$$

such that $F(\bm L)$ is a sheaf quantization of $\bm L$.

These conjectures propose a comparison between the sheaf-theoretic and Fukaya-theoretic constructions for Liouville manifolds. In the exact case, the assertion has been proved by Viterbo, while the stated general Novikov-coefficient version is resolved according to the supplied status evidence.