restriction to $({\mathcal O}_K^\times)^n$). We say that a point $\delta\in \mathcal{W}^n_L$ is [*algebraic*]{} if $\delta=\delta_{\bf k}$ for some ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. We say that an algebraic $\delta=\delta_{\bf k}$ is [*dominant*]{} (resp. [*strictly dominant*]{}) if moreover $k_{\tau,i}\geq k_{\tau,i+1}$ (resp. $k_{\tau,i}> k_{\tau,i+1}$) for $i\in \{1,\dots,n-1\}$ and $\tau\in {\mathrm{Hom}}(K,L)$.

We write ${\mathcal{T}}_{\rm reg}\subset {\mathcal{T}}_L$ for the Zariski-open complement of the $L$-valued points $z^{-\bf k}, \vert z\vert_K z^{{\bf k}+{\bf 1}}$, with ${\bf k}=(k_\tau)_{\tau:K\hookrightarrow L}\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$. We write ${\mathcal{T}}_{\rm reg}^n$ for the Zariski-open subset of characters $(\delta_1,\dots,\delta_n)$ such that $\delta_i/\delta_j\in {\mathcal{T}}_{\rm reg}$ for $i\neq j$.

We fix a continuous representation ${\overline{r}}:\mathcal{G}_K\rightarrow {\mathrm{GL}}_n(k_L)$ and let $R_{{\overline{r}}}^{\square}$ be the framed local deformation ring of ${\overline{r}}$ (a local complete noetherian $\mathcal{O}_L$-algebra of residue field $k_L$). We write $\mathfrak{X}_{{\overline{r}}}^\square:=({\mathrm{Spf}}\, R_{{\overline{r}}}^\square)^{{\mathrm{rig}}}$ for the rigid analytic space over $L$ associated to the formal scheme ${\mathrm{Spf}}\, R_{{\overline{r}}}^\square$. Recall that a representation $r$ of ${\mathcal{G}}_K$ on a finite dimension $L$-vector space is called *trianguline of parameter* $\delta=(\delta_1,\dots,\delta_n)$ if the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ over ${\mathcal{R}}_{L,K}$ associated to $r$ admits an increasing filtration ${\mathrm{Fil}}_\bullet$ by sub-$(\varphi,\Gamma_K)$-modules over ${\mathcal{R}}_{L,K}$ such that the graded piece ${\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}$ is isomorphic to ${\mathcal{R}}_{L,K}(\delta_i)$. We let $X_{\rm tri}^\square({\overline{r}})$ be the associated framed trianguline variety, see [@BHS §2.2] and 