Integrality conjecture for the true mirror map of Fano data

Let $K_{ij}$ be the monoid defined by allowing the $(i,j)$-coordinate to be negative subject to the stated sum condition, let $\tau_{ij}$ be the associated formal series, and define the true mirror map by

$$\psi_{ij}^{\mathrm{t}}=\exp\left((\phi_{ij}+\tau_{ij})/\phi_0\right)\in\mathbb{Q}[[K_0+K_{ij}]].$$

Assume $\boldsymbol{v}_{ij}\neq 0$. True mirror-map integrality conjecture. If the vectors $\boldsymbol{v}_{ij}$ are Fano, then $\psi_{ij}^{\mathrm{t}}$ has integer coefficients for every $(i,j)\in I$. This is the paper's second conjecture, concerning the true rather than naive mirror map; the claim is stated without a resolution in the supplied text.