Let $X$ be a toric CY 3-fold, and this will be our A-side.

Let $N_X$ and $M_X$ be 1PS and character lattice. $\Sigma_X$ lives in $N_X$. Let $P_\Sigma$ denote the compact polytope, being the convex hull of $0$ and the ray generators of $\Sigma$.

Let $Y$ be the toric variety with moment polytope $P_\Sigma$. Let $Y'$ be the toric boundary divisor of $Y$ corresponding to the top face of $P_\Sigma$.

I am reading this paper by Zaslow-Treumann-Williams, https://arxiv.org/pdf/1810.05985.pdf But it does not discuss the relation with toric CY 3fold. Then, let me read this physics paper: FHKV, Dimer models from mirror symmetry and quivering amoebæ.

A toric CY $X$, with fan $\Sigma$; is mirror to another CY (ha? not a LG model?) $$ Y = \{uv = P(z,w) = \sum_{\alpha \in Q_\Sigma} c_\alpha z^\alpha \} $$ So, $Y$ is not toric. Why we want to consider the conic fibration over this 'spectral curve'?