The Hyperplane Conjecture for periods of anticanonical hypersurfaces

Let $X_{\underline{t}}$ be the family of anticanonical hypersurfaces associated with the polytope $\Delta$, let $\nabla$ be its Gauss–Manin connection, and let $\mathscr{B}_{\Delta}$ be the cohomology-valued series defined from the Mori cone. Write $X^{\circ}\subset \mathbb{P}_{\Delta^{\circ}}$ for an anticanonical hypersurface. Hyperplane Conjecture. The periods of $\nabla$-flat sections of $H^{n-1}(X_{\underline{t}})$ are the $\mathbb{C}$-linear combinations of coefficients of cohomology classes in $\mathscr{B}_{\Delta}\cup [X^{\circ}]$. In the examples where $\mathbb{P}_{\Delta^{\circ}}=\mathbb{P}^2$, this gives the actual periods of $H^1(E_t)$; the conjecture concerns isolating the highest-weight part of the mixed Hodge structure among the GKZ solutions.