Percolation with random one-dimensional reinforcements

We study inhomogeneous Bernoulli bond percolation on the graph $G\times \mathbb{Z}$, where $G$ is a connected quasi-transitive graph. The inhomogeneity is introduced through a random region $R$ around the origin axis $\{0\}\times \mathbb{Z}$, where each edge in $R$ is open with probability $q$ and all other edges are open with probability $p$. When the region $R$ is defined by stacking or overlapping boxes with random radii centered along the origin axis, we derive conditions on the moments of the radii, based on the growth properties of $G$, so that for any subcritical $p$ and any $q<1$, the non-percolative phase persists.