Fewer edges and phase transition in the long-range contact process

Consider the long-range contact process (LoRaC) on the spatial model of the paper, with edge length $|y-x|$ and a parameter $\alpha>0$. Fewer edges. The LoRaC has a phase transition even if edges are present only with probability

$$\exp(-\alpha|y-x|).$$

The surrounding discussion identifies this as the unresolved exponential-decay case: infinitely long edges are required for percolation in the bounded-length setting, but the precise effect of exponential decay is not known.