Pemantle–Stacey conjecture on the invariance of the contact-process critical rate under edge addition

Let $G=(V,E)$ and $G'=(V,E')$ be connected graphs with the same vertex set $V$, where

$$E'=E\cup\{\{x,y\}\}$$

for some $x,y\in V$. Write $\lambda_c(G)$ for the critical infection rate of the contact process on $G$. Pemantle–Stacey conjecture. The critical rates are equal:

$$\lambda_c(G)=\lambda_c(G').$$

The conjecture asserts that adding a single edge does not affect the survival–extinction threshold of the contact process. Proving it in full generality remains an open problem.