Extinction due to sparse environment in the long-range contact process

Let $q^{(t)},p\in(0,1)$ be given and let $\alpha>1$. Let $q^{(x)}$ denote the parameter controlling the sparse spatial environment in the long-range contact process (LoRaC). Extinction due to sparse environment. There exists $q_c^{(x)}\in(0,1)$ such that, for every $q^{(x)}>q_c^{(x)}$, the LoRaC does not percolate. The conjecture concerns whether sufficiently sparse environments prevent survival; the paper explains that long lockdowns yield extinction by comparison with a Galton--Watson process, but that the effect of the sparse environment is more difficult and remains open.