The coherent-constructible correspondence for smooth complete fans

Let $M$ be a free abelian group of rank $n$, let $N$ be its dual, and let $\Sigma$ be a smooth complete fan in $N_\bR:=N\otimes_\bZ\bR$. Let $X_\Sigma$ be the associated smooth toric variety, set $T^n:=M_\bR/M$, and define

$$\Lambda_\Sigma:=\bigcup_{\sigma\in\Sigma}p(\sigma^\perp)\times(-\sigma)\subset T^*T^n,$$

where $p:M_\bR\to T^n$ is the quotient map. Let $\kappa_\Sigma:\operatorname{Coh}(X_\Sigma)\to\operatorname{Sh}^c_{\Lambda_\Sigma}(T^n)$ be the fully faithful morphism to the dg category of constructible sheaves whose microsupports are contained in $\Lambda_\Sigma$. The coherent-constructible correspondence. The morphism $\kappa_\Sigma$ induces a quasi-equivalence of dg categories.

This is the coherent-constructible correspondence formulated by Fang–Liu–Treumann–Zaslow; its equivariant version was proved by them. The claim is known when $\Sigma$ is cragged or has dimension $2$, while the general smooth complete case remains open.