Vlasenko's p-adic Frobenius and connection matrix conjecture

Let $f$ be a homogeneous polynomial defining a smooth hypersurface family, and let $\alpha_s$ be the matrices associated with the Laurent polynomial construction, with $\sigma$ the Frobenius lift and $D$ a derivation on the base. Under the stated Hodge-theoretic and Hasse–Witt assumptions, the unit-root $F$-crystal $U_0$ has Frobenius matrix $F$ and connection matrix $\nabla_D$ in the indicated basis. Vlasenko's conjecture. The Frobenius matrix is the $p$-adic limit

$$F=\lim_{s\to \infty} \alpha_{s+1}\sigma(\alpha_s)^{-1},$$

and the connection matrix is

$$\nabla_D=\lim_{s\to \infty} D(\alpha_s)(\alpha_s)^{-1}.$$

These limits are conjectured to describe the Frobenius and connection on the unit-root part of crystalline cohomology; the paper applies Katz's method to prove the relevant conjecture in the hypersurface setting, while the general formulation depends on the stated hypotheses.