Tropical coamoeba realization and equivariant homological mirror symmetry

Let $\Delta$ be a convex lattice polytope in $\bR^n$ containing the origin in its interior. Let $X$ be the toric Fano stack associated with $\Delta$, let $W:(\bC^*)^n\to\bC$ be a Laurent polynomial whose Newton polytope is $\Delta$, let $G$ be a tropical coamoeba of $W$, and let $\scA_{\Gtilde}$ be the associated $A_\infty$-category. Tropical coamoeba conjecture. There is a choice of $W$ and $G$ such that

$$\scA_{\Gtilde} \cong \Fuk \Wtilde$$

and

$$D^b\scA_{\Gtilde} \cong D^b \operatorname{Coh}^{\bT}(X).$$

These two assertions split equivariant homological mirror symmetry into a Fukaya-theoretic tropical-coamoeba comparison and an algebraic comparison with equivariant coherent sheaves. The source presents them as the two steps of its program; the paper establishes the relevant statements for the projective-space case.