Homological Mirror Symmetry for Conic Bundle

We study the homological mirror symmetry statement where A-side is the conic bundle Hori--Vafa mirror $\mathcal{Y} = \{uv = f(z)\} \subset \mathbb{C}^2 \times (\mathbb{C}^\ast)^n$ for a Laurent polynomial $f$ in $(\mathbb{C}^\ast)^n$, and B-side is some a toric Calabi--Yau $(n+2)$-fold with a smooth anti-canonical divisor removed $\mathcal{X}^\circ = \mathcal{X} \setminus w^{-1}(-1)$. We show that when $\mathcal{X}$ is the canonical bundle of a toric Fano $n$-orbifold $S$ and $f$ is its Givental superpotential, the strong deformation retraction skeleton $\mathsf{L}$ of $\mathcal{Y}$ in the sense of RSTZ (Ruddat--Sibilla--Treumann--Zaslow in Geom. Topol. 18(3):1343--1395, 2014) has a Weinstein neighborhood $U$, such that the wrapped microlocal sheaf category $\mu\mathrm{Sh}^w_{\mathsf{L}}(\mathsf{L}) \cong \mathrm{Coh}(\mathcal{X}^\circ)$. This proves a microlocal categorical version of the SYZ mirror in (Abouzaid--Auroux--Katzarkov in Publ. math. IH\'ES 123(1):199--282, 2016, Thm. 1.7). We also extend the definition of characteristic cycles for constructible sheaves in cotangent bundles from (Kashiwara--Schapira in Sheaves on Manifolds, Grundlehren math. Wiss. 292, Springer, 1990, Ch. IX) to finite-rank objects in $\mu\mathrm{Sh}^w_{\mathsf{L}}(\mathsf{L})$, and describe the characteristic cycles for objects mirror to a coherent sheaf supported on $S$.