Ising influence bounded by hyperbolic tangent

Let $d\ge 1$, let spins be $s_i\in \{-1,+1\}$, let $h_i\in \mathbb{R}$, and let $J_{ij}=J_{ji}\in \mathbb{R}$ with $J_{ii}=0$ and $J_{ij}=0$ whenever $\mathrm{\parallel }i-j{\mathrm{\parallel }}_{\mathrm{\infty }}>R$ for some finite integer $R$. No translation invariance is assumed. For finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$ and exterior $\eta \in \{-1,+1{\}}^{{\mathrm{\Lambda }}^c}$, define ${\widehat{s}}_i=s_i$ on $\mathrm{\Lambda }$ and ${\widehat{s}}_i={\eta }_i$ outside, and use $E_{\mathrm{\Lambda }}(s_{\mathrm{\Lambda }}\mid \eta )=-{\sum }_{\{i,j\}:\{i,j\}\cap \mathrm{\Lambda }\mathrm{\ne }\mathrm{\varnothing }}{J}_{ij}{\widehat{s}}_i{\widehat{s}}_j-{\sum }_{i\in \mathrm{\Lambda }}{h}_i{s}_i,$ where the pair sum is over unordered distinct pairs and is finite by the range assumption. For site $i$ and exterior spins $\eta \in \{-1,+1{\}}^{{\mathbb{Z}}^d\setminus \{i\}}$, the one-site law is explicitly ${\gamma }_i(s\mid \eta )=\frac{\exp \text{\hspace{0.17em}⁣}\left(\beta [h_{i}s+{\sum }_{k\mathrm{\ne }i}{J}_{ik}s{\eta }_k]\right)}{{\sum }_{u\in \{-1,+1\}}\exp \text{\hspace{0.17em}⁣}\left(\beta [h_{i}u+{\sum }_{k\mathrm{\ne }i}{J}_{ik}u{\eta }_k]\right)},s\in \{-1,+1\};$ the sums are finite by the range condition. For $i\mathrm{\ne }j$, define $c_{ij}:={\sup }_{\eta ,{\eta }^{\mathrm{\prime }}:{\eta }_k={\eta }_k^{\mathrm{\prime }}(k\mathrm{\ne }j)}\frac{1}{2}{\sum }_{s\in \{-1,+1\}}\mathrm{\mid }{\gamma }_i(s\mid \eta )-{\gamma }_i(s\mid {\eta }^{\mathrm{\prime }})\mathrm{\mid },c_{ii}:=0.$ Then $c_{ij}=0$ if $J_{ij}=0$, and otherwise $c_{ij}\le \tanh (\beta \mathrm{\mid }{J}_{ij}\mathrm{\mid })\mathrm{.}$ In particular, ${\sup }_i{\sum }_j\tanh (\beta \mathrm{\mid }{J}_{ij}\mathrm{\mid })<1$ is a sufficient Dobrushin row-sum bound for every bounded field profile $(h_i)$.