Let $X$ be a nonsingular complex toric variety with moment map $\mu : X \to P$ over a convex polytope $P$. Given a facet $F$ of $P$, its preimage $\mu^{-1}(F)$ is a complex codimension 1 subvariety of $X$, so by Poincare duality, induces a cohomological class $\mathfrak{z}_F \in H^2(X; \mathbb{C})$.

Is $\mathfrak{z}_F$ integral, i.e. does it lie in the image of the map $H^2(X; \mathbb{Z}) \to H^2(X; \mathbb{C})$ induced by the inclusion of coefficient rings $\mathbb{Z} \hookrightarrow \mathbb{C}$?