Local survival is equivalent to metastability for the contact process

Let $I_n(t)$ denote the expected number of infected vertices at time $t$ for the contact process on the dynamical graph with initially all vertices infected. Write $\theta(\lambda)$ for the probability that the contact process with infection rate $\lambda$ survives on the $GSMPGW(\kappa,\beta)$, starting from only the root infected. The process is said to have slow extinction when its mean extinction time is exponential in the number of vertices, and it has metastability when, for some $\varepsilon>0$, its expected number of infected vertices remains asymptotically bounded away from zero and approximately constant at times tending to infinity slower than $e^{\varepsilon n}$. Local survival is equivalent to metastability. We have $\theta(\lambda)>0$ precisely when there is slow extinction of the contact process, and in that case there is metastability, with metastable density $\theta(\lambda)$. The theorem preceding this conjecture establishes that, in the slow extinction regime, the metastable density equals $\theta(\lambda)>0$; the conjectured equivalence beyond that regime is not proved in the source.