Fano data conjecture on positivity and integrality of the naive mirror map

Let $I$ index vectors $\boldsymbol{v}_{ij}$, let $K_0$ be the monoid used to define the formal power series $\boldsymbol{z}^{\boldsymbol{k}}$, and let $\boldsymbol{v}_{ij}$ be Fano when the origin is the unique interior lattice point of the associated polytope. Define

$$\psi_{ij}^{\mathrm{n}}(\boldsymbol{z})=\exp(\phi_{ij}/\phi_0),$$

where $\phi_0$ and $\phi_{ij}$ are the formal series defined above. Fano data conjecture. If $(\boldsymbol{v}_{ij})$ is Fano, then for all $(i,j)\in I$, $\psi_{ij}^{\mathrm{n}}$ has integer coefficients, so that it lies in $\mathbb{Z}[[K_0]]$, and

$$\log\psi_{ij}^{\mathrm{n}}=\phi_{ij}/\phi_0$$

has non-negative coefficients, so that it lies in $\mathbb{Q}_{\geq 0}[[K_0]]$. Together these assertions imply $\psi_{ij}^{\mathrm{n}}\in\mathbb{N}[[K_0]]$. This conjecture predicts positivity and integrality properties of the naive mirror map; the paper notes that in the quintic example it recovers the corresponding expected properties of the quintic mirror map.