The Fukaya-to-sheaf quantization conjecture for cotangent bundles

Let $\bK$ be a field, let $M$ be a real manifold, and let $\bR_t$ be the real line with coordinate $t$. Write $\mathrm{Sh}_{\tau>0}^{\bR^\delta}(M\times \bR_t)$ for the equivariant Tamarkin category defined by quotienting by objects with microsupport in $\{\tau\leq 0\}$, where $\tau$ is the cotangent coordinate on $\bR_t$. A sheaf quantization of a Lagrangian $L\subset T^*M$ is an object whose positive microsupport projects to $L$. Fukaya-to-sheaf quantization conjecture. There exists an infinitesimally wrapped Fukaya category $Fuk(T^*M)$ of nonexact Lagrangians in $T^*M$ with a natural embedding

$$Fuk(T^*M)\hookrightarrow \mathrm{Sh}_{\tau> 0}^{\bR^\delta}(M\times \bR_t).$$

The conjecture is known for integral Lagrangians; the general assertion for nonexact Lagrangians remains open. In the paper it is used to predict sheaf quantizability of generic rotations of holomorphic Lagrangians.