The Hasse–Witt invariant and mirror-map expansion conjecture

Let $X_z$ be a family of Calabi–Yau $n$-folds defined over a perfect field of characteristic $p$, let $\alpha_z$ be a holomorphic $n$-form, and let $\operatorname{HP}(X_z,\alpha_z)$ be the normalized holomorphic period. Write $\operatorname{HW}(X_z,\alpha_z)$ for the scalar determined by the Cartier operator and let $A_p(z):=\operatorname{HP}(X_z,\alpha_z)^{p-1}\operatorname{HW}(X_z,\alpha_z)$. Hasse–Witt and mirror-map conjecture. For all primes $p$ not dividing $N$, the Hasse–Witt invariant $\operatorname{HW}(X_z,\alpha_z)$, up to sign, is the truncation of the holomorphic period at degree $p-1$, and $A_p(z(q))$ has $p$-integral coefficients and satisfies

$$A_p(z(q))\equiv_p1$$

up to sign. The claim is motivated by computations and examples; the text later records partial results and says that the required $p$-integrality remains unresolved in general.