The Torelli conjecture for mirror Landau–Ginzburg models

Let $X_1$ and $X_2$ be Fano varieties with mirror Landau–Ginzburg models $(X_1^\vee,W_1)$ and $(X_2^\vee,W_2)$. Write $\pi_{W_i}(t)$ for the classical period of the model, and let $\widehat{G}_{X_i}(t)$ denote the corresponding regularized quantum period. The Torelli conjecture. If

$$\pi_{W_1}(t)=\pi_{W_2}(t)$$

(or equivalently if $\widehat{G}_{X_1}(t)=\widehat{G}_{X_2}(t)$), then $X_1^\vee$ and $X_2^\vee$ are isomorphic. This reformulates the period conjecture as a Torelli statement: the period should determine the mirror Landau–Ginzburg model, although the claim is not established in the stated generality.