Apparently this issue was worked out shortly after, during Go Yamashita's stay at the IHES in 2006, and after some discussion with Deligne.

For any Tate motive $M$ unramified at $v$, its crystalline realization is defined as $D_\mathrm{crys}(M_p)$, where $D_\mathrm{crys}$ is Fontaine's functor $(B_\mathrm{crys} \otimes_{\mathbb{Q}_p}-)^{G_{k_v}}$, and $M_p$ is the p-adic realization of $M$.

This realization is functorial, and has the comparison isomorphism:

$$k_v \otimes_{k_0,v}D_\mathrm{crys}(M_p) \cong k_v\otimes_k M_\mathrm{dR}$$

All of this is perfectly explained and proved in section 4 of:

• Go Yamashita, Bounds for the Dimensions of p-adic Multiple L-Value Spaces (2009)

Apparently this issue was worked out shortly after, during Go Yamashita's stay at the IHES in 2006, and after some discussion with Deligne.

For any Tate motive $M$ unramified at $v$, its crystalline realization is defined as $D_\mathrm{crys}(M_p)$, where $D_\mathrm{crys}$ is Fontaine's functor $(B_\mathrm{crys} \otimes_{\mathbb{Q}_p}-)^{G_{k_v}}$, and $M_p$ is the p-adic realization of $M$.

This realization is functorial, and has the comparison isomorphism:

$$k_v \otimes_{k_0,v}D_\mathrm{crys}(M_p) \cong k_v\otimes_k M_\mathrm{dR}$$

All of this is perfectly explained and proved in section 4 of:

• Go Yamashita, Bounds for the Dimensions of p-adic Multiple L-Value Spaces (2009)