Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?

For example: If $K\ne \mathbb{C} $ and $X\rightarrow \operatorname{spec}(K)$ be a K3-surface. We do not have a Hodge isometry or a Period map (as far as I know). However we have etale cohomology and crystalline cohomology. If $K$ is a local field, then Falting's theorem on $p$-adic Hodge theory give us a Hodge-like decomposition of $$H^2_{et}(X_{K^{al}},\mathbb{Z}_p) \otimes \mathbb{C}_p \longrightarrow H^0(X,\Omega^2_X)(-2)\oplus H^1(X,\Omega^1_X)(-1)\oplus H^2(X,\Omega^0_X)(0)$$ as $G$-representations for $G$ the Galois group of $K^{al} /K$.

The theorem I am looking for is something kind of if we have a map $$\phi: H^2_{et}(X_{K^{al}},\mathbb{Z}_p) \longrightarrow H^2_{et}(X'_{K^{al}},\mathbb{Z}_p)$$ wich preserves the Hodge-like decomposition after tensoring with a Fontain's Ring (like $B_{dR}$), then there exists an isomorphism $f:X\to X'$ inducing $\phi$ on etale cohomology.

Any idea of a work/paper on this direction?