Weak p-adic Grothendieck period conjecture

Let $X$ be a variety over $\overline{\mathbb{Q}}$ with good reduction $X_p$ at a prime $p$. Let $Z_0$ be the $\mathbb{Q}$-vector space of algebraic classes on $X$, let $Z_p$ be the $\mathbb{Q}$-vector space of algebraic classes on $X_p$, and use the comparison embedding into de Rham cohomology. Weak p-adic Grothendieck period conjecture. Is the inclusion

$$Z_0\subset Z_p\cap H^*_{\mathrm{dR}}(X,\overline{\mathbb{Q}})$$

an equality? The paper states that the analogous conjecture is false in general in the classical setting, while the p-adic question is posed here as an open problem.