It is stated here (the remarks following equation $(2)$) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ of $\mathbb{Q}_p$ are not $\mathbb{C}_p$-admissible. In fact, $\mathbb{C}_p\otimes_{\mathbb{Q}_p}H_{ét}^r(X_{\overline{K}},\mathbb{Q}_p)$ is isomorphic to the graded module of $\mathbb{C}_p\otimes_K H_{\rm dR}^r(X)$ for the de Rham filtration, but not to $\mathbb{C}_p\otimes_K H_{\rm dR}^r(X)$ itself. Is there a reference for this result or a quick proof using a characterization of $\mathbb{C}_p$-admissibility (at least to prove that they are not $\mathbb{C}_p$-admissible) ?

It is stated in Caruso - An introduction to $p$-adic period rings (the remarks following equation (2)) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ of $\mathbb{Q}_p$ are not $\mathbb{C}_p$-admissible. In fact, $\mathbb{C}_p\otimes_{\mathbb{Q}_p}H_\text{ét}^r(X_{\overline{K}},\mathbb{Q}_p)$ is isomorphic to the graded module of $\mathbb{C}_p\otimes_K H_{\text{dR}}^r(X)$ for the de Rham filtration, but not to $\mathbb{C}_p\otimes_K H_{\text{dR}}^r(X)$ itself. Is there a reference for this result or a quick proof using a characterization of $\mathbb{C}_p$-admissibility (at least to prove that they are not $\mathbb{C}_p$-admissible) ?