The specialization conjecture for rigid and crystalline cohomology

Let $X\to S$ be proper, flat and generically smooth, with special fibre $X_s$ over $k$ and geometric generic fibre $X_{\bar\eta}$. Let $H^i_{\operatorname{rig}}(X_s/k)$ be rigid cohomology, let $D_{\operatorname{cris}}(H^i(X_{\bar\eta},\mathbb Q_p))$ be the crystalline period module, and let $\hat{\mathbb Q}_p^{\operatorname{ur}}$ be the $p$-adic completion of the maximal unramified extension of $\mathbb Q_p$. Specialization conjecture. There is a $\phi$-equivariant specialization map

$$H^i_{\operatorname{rig}}(X_s/k)\xrightarrow{\operatorname{sp}'}D_{\operatorname{cris}}(H^i(X_{\bar\eta},\mathbb Q_p))$$

and a commutative diagram relating it to the étale specialization map, with vertical maps inducing

$$\lambda_s:H^i(X_{\bar s},\mathbb Q_p)\cong\bigl(H^i_{\operatorname{rig}}(X_s/k)\otimes_{\mathbb Q_p}\hat{\mathbb Q}_p^{\operatorname{ur}}\bigr)^{\phi\otimes\phi=1}\cong H^i_{\operatorname{rig}}(X_s/k)^{\operatorname{slope}0}$$

and

$$\lambda_\eta:H^i(X_{\bar\eta},\mathbb Q_p)^I\cong\bigl(D_{\operatorname{cris}}(H^i(X_{\bar\eta},\mathbb Q_p))\otimes_{\mathbb Q_p}\hat{\mathbb Q}_p^{\operatorname{ur}}\bigr)^{\phi\otimes\phi=1}\cong D_{\operatorname{cris}}(H^i(X_{\bar\eta},\mathbb Q_p))^{\operatorname{slope}0}.$$

This is the $p$-adic analogue of the corresponding $l$-adic specialization statement and is presented as conjectural.