The Fermat mirror-map conjecture

For each positive integer $n$, let $\mathscr{F}_n$ be the Fermat-type Calabi–Yau $n$-fold and let $t$ denote the mirror map for its Fermat pencil, with Fermat point $\psi=0$. Mirror-map conjecture. The value of the mirror map at the Fermat point has the form

$$t\big|_{\psi=0}=\frac{1}{2}+\xi i,$$

where $\xi$ is a real algebraic number. The numerical examples in the paper support this form, and the authors note that $\xi$ depends on $n+2$; no general proof is given.