Approximation on slabs and uniqueness for inhomogeneous percolation with a plane of defects

Let $\mathbb{L}^{d} = ( \mathbb{Z}^{d},\mathbb{E}^{d} )$ be the $d$-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $\mathbb{L}^{d}$ in which every edge inside the $s$-dimensional hyperplane $\mathbb{Z}^{s} \times \{ 0 \}^{d-s}$, $2 \leq s < d$, is open with probability $q$ and every other edge is open with probability $p$. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $p \neq p_{c}(d)$, where $p_{c}(d)$ denotes the threshold for homogeneous percolation, and that the critical point $(p,q_{c}(p))$ can be approximated on the phase space by the critical points of slabs, for any $p < p_{c}(d)$.