Let $E$ be a separable $\mathbb R$-Banach space, $\rho_r$ be a metric on $E$ for $r\in(0,1]$ with $\rho_r\le\rho_s$ for all $0<r\le s\le1$, $\rho:=\rho_1$, $$d_{r,\:\delta,\:\beta}:=1\wedge\frac{\rho_r}\delta+\beta\rho\;\;\;\text{for }(r,\delta,\beta)\in[0,1]\times(0,\infty)\times[0,\infty)$$ and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$.

Assume we arbe able to show that for all $n\in\mathbb N$ there is a $\alpha\in[0,1)$ and $(r,\delta,\beta)\in[0,1]\times(0,\infty)\times(0,1)$ with$^1$ $$\operatorname W_{d_{r,\:\delta,\:\beta}}\left(\delta_x\kappa_n,\delta_y\kappa_n\right)\le\alpha\operatorname W_{d_{r,\:\delta,\:\beta}}\left(\delta_x,\delta_y\right)\tag1$$ for all $x,y\in E$, where $\delta_x$ denotes the Dirac measure on $(E,\mathcal B(E))$ at $x\in E$. Why are we able to conclude that there is a $(c,\lambda\in[0,\infty)^2$ with $$\operatorname W_\rho\left(\nu_1\kappa_t,\nu_2\kappa_t\right)\le ce^{-\lambda t}\operatorname W_\rho\left(\nu_1,\nu_2\right)\tag2$$ for all $\nu_1,\nu_2\in\mathcal M_1(E)$ and $t\ge0$?

It's clear to me that if $\kappa$ is any Markov kernel on $(E,\mathcal B(E))$ and $d$ is any metric on $E$ such that there is a $\alpha\ge0$ with $\operatorname W_d\left(\delta_x\kappa,\delta_y\kappa\right)\le\alpha\operatorname W_d\left(\delta_x,\delta_y\right)$ for all $x,y\in E$, then this extends to $\operatorname W_d(\mu\kappa,\nu\kappa)\le\alpha\operatorname W_d(\mu,\nu)$ for all $\mu,\nu\in\mathcal M_1(E)$. Moreover, it's clear that $\operatorname W_d\left(\delta_x,\delta_y\right)=d(x,y)$.

Note that for any choice of $(r,\delta,\beta)\in[0,1]\times(0,\infty)\times[0,\infty)$, it holds $$\beta\rho\le d_{r,\:\delta,\:\beta}\le\left(\frac1\delta+\beta\right)\rho.\tag3$$

*Remark*: The desired claim seems to be used in the proof of Theorem 3.4 in https://arxiv.org/pdf/math/0602479.pdf.

$^1$ If $(E,d)$ is a complete separable metric space and $\mathcal M_1(E)$ is the space of probability measures on $\mathcal B(E)$, then the Wasserstein metric $\operatorname W_d$ on $\mathcal M_1(E)$ satisfies the identity $$\operatorname W_d(\mu,\nu)=\sup_{\substack{f\::\:E\:\to\:\mathbb R\\|f|_{\operatorname{Lip}(d)}\:\le\:1}}(\mu-\nu)f\;\;\;\text{or all }\mu,\nu\in\mathcal M_1(E),$$ where $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:E\\x\:\ne\:y}}\frac{|f(x)-f(y)|}{d(x,y)}\;\;\;\text{for }f:E\to\mathbb R$$ and $\mu f:=\int f\:{\rm d}\mu$ for $\mu$-integrable $f:E\to\mathbb R$.