[@HellmannFS]. Recall that $X_{\rm tri}^\square({\overline{r}})$ is the reduced rigid analytic space over $L$ which is the Zariski closure in $\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ of: $$\label{ucris}
U_{\rm tri}^\square({\overline{r}}):=\{{\rm points}\ (r,\delta)\ {\rm in}\ {\mathfrak{X}}^\square_{{\overline{r}}}\times{\mathcal{T}}^n_{\rm reg}\ {\rm such\ that}\ r\ \text{is trianguline of parameter}\ \delta\}$$ (the space $U_{\rm tri}^\square({\overline{r}})$ is denoted $U_{\rm tri}^\square({\overline{r}})^{\rm reg}$ in [@BHS §2.2]). The rigid space $X_{\rm tri}^\square({\overline{r}})$ is reduced equidimensional of dimension $n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}$ and its subset $U_{\rm tri}^\square({\overline{r}})\subset X_{\rm tri}^\square({\overline{r}})$ turns out to be Zariski-open, see [@BHS Th.2.6]. Moreover by [*loc. cit.*]{} the rigid variety $U_{\rm tri}^\square({\overline{r}})$ is smooth over $L$ and equidimensional, hence there is a bijection between the set of connected components of $U_{\rm tri}^\square({\overline{r}})$ and the set of irreducible components of $X_{\rm tri}^\square({\overline{r}})$.

We denote by $\omega:X_{\rm tri}^\square({\overline{r}})\rightarrow \mathcal{W}^n_L$ the composition $X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\twoheadrightarrow \mathcal{T}^n_L \twoheadrightarrow \mathcal{W}^n_L$. If $x$ is a point of $X_{\rm tri}^\square({\overline{r}})$, we write $x=(r,\delta)$ where $r\in \mathfrak{X}_{{\overline{r}}}^\square$ and $\delta=(\delta_1,\dots,\delta_n)\in \mathcal{T}_L^n$. We say that a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is *crystalline* if $r$ is a crystalline representation of $\mathcal{G}_K$.

\[paramofcrystpt\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline point. Then for $i\in \{1,\dots,n\}$ there exist ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ and $\varphi_i\in k(x)^\times$ such that: $$\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i).$$ Moreover the $(k_{\tau,i})_{i,\tau}$ are the labelled Hodge-Tate 