weights of $r$ and the $\varphi_i$ are the eigenvalues of the geometric Frobenius on the (unramified) Weil-Deligne representation ${\rm WD}(r)$ associated to $r$ (cf. [@Fo]).

The fact that the $(k_{\tau,i})_{i,\tau}$ are the Hodge-Tate weights of $r$ follows for instance from [@BHS Prop.2.9]. By [@KPX Th.6.3.13] there exist for each $i$ a continuous character $\delta'_i:K^\times \rightarrow k(x)^\times$ such that $r$ is trianguline of parameter $\delta':=(\delta'_1,\dots,\delta'_n)$ and such that $\delta_i/\delta'_i$ is an algebraic character of $K^\times$ (i.e. of the form $z^{{\bf k}}$ for some ${\bf k}\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$). It thus suffices to prove that each $\delta'_i$ is of the form $z^{{\bf k}'_i}{\rm unr}(\varphi_i)$ for some ${\bf k}'_i\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$ where the $\varphi_i\in k(x)^\times$ are the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$, or equivalently (using the definition of ${\rm WD}(r)$) are the eigenvalues of the linearized Frobenius $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on the $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-module $D_{\rm cris}(r):=(B_{\rm cris}\otimes_{{\mathbb{Q}}_p}r)^{\mathcal{G}_K}$. By [@Berger Th.3.6] there is an isomorphism (recall $t$ is “Fontaine’s $2i\pi$”): $$\begin{aligned}
\label{berger}
D_{\rm cris}(r)\cong D_{{\mathrm{rig}}}(r)[\tfrac{1}{t}]^{\Gamma_K},\end{aligned}$$ and a triangulation ${\mathrm{Fil}}_\bullet$ of $D_{{\mathrm{rig}}}(r)$ with graded pieces giving the parameter $\delta'$ induces a complete $\varphi$-stable filtration ${\mathcal{F}}_\bullet$ on $D_{\rm cris}(r)$ such that ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ is the filtered $\varphi$-module associated to ${\mathcal{R}}_{L,K}(\delta'_i)={\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}$ by the same recipee as (\[berger\]). It follows from this and from 