Truncated Connectivities in a highly supercritical anisotropic percolation model

We consider an anisotropic bond percolation model on ${\mathbb{Z}}^2$, with $\mathbf{\text{p}}=(p_h,p_v)\in [0,1{]}^2$, $p_v>p_h$, and declare each horizontal (respectively vertical) edge of ${\mathbb{Z}}^2$ to be open with probability $p_h$(respectively $p_v$), and otherwise closed, independently of all other edges. Let $x=(x_1,x_2)\in {\mathbb{Z}}^2$ with $0<x_1<x_2$, and $x^{\mathrm{\prime }}=(x_2,x_1)\in {\mathbb{Z}}^2$. It is natural to ask how the two point connectivity function $\text{prob}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\text{prob}(\{0\leftrightarrow x\})>\text{prob}(\{0\leftrightarrow x^{\mathrm{\prime }}\})$. In this note we give an affirmative answer in the highly supercritical regime.