Häggström–Pemantle coexistence conjecture for the two-type Richardson model

Let $G$ denote the event of infinite coexistence in the two-type Richardson model on $\mathbb{Z}^d$, with infection intensities normalized to $1$ and $\lambda$ for the two types, respectively. For $d e 2$, the model is started from the fertile initial configuration $S_0^1=\mathbf{0}$ and $S_0^2=\mathbf{1}$, and $P^{1,\lambda}$ denotes its law.

Häggström–Pemantle's conjecture. In every dimension $d\geq 2$, infinite coexistence has positive probability if and only if the infection intensities are equal:

$$P^{1,\lambda}(G)>0 \quad\Longleftrightarrow\quad \lambda=1.$$

The statement is the proposed characterization of coexistence, and the supplied text records that the $\lambda=1$ direction has been proved for every $d\geq 2$; the only-if direction remains the conjectural part.