Coexistence conjecture for two-type growth models

Let $d\geq 2$. For $i=1,2$, let $A_i$ be the event that infection type $i$ reaches sites arbitrarily far from the origin, and let $A=A_1\cap A_2$ be the coexistence event. Let $\tilde{P}^{\lambda_1,\lambda_2}$ denote the law of the discrete two-type process started from single infections at the origin and at $(1,0,\ldots,0)$, and let $P^{\lambda_1,\lambda_2}$ denote the law of the continuum two-type process started from unit balls centered at the origin and at $(2,0,\ldots,0)$. Here $\lambda_1$ and $\lambda_2$ are the infection intensities, and $F$ is the outburst-radius distribution. Coexistence conjecture. For every dimension $d\geq 2$, both of the following hold:

$$\tilde{P}^{\lambda_1,\lambda_2}(A)>0\quad\Longleftrightarrow\quad \lambda_1=\lambda_2;$$

and, if $F$ satisfies the stated condition,

$$P^{\lambda_1,\lambda_2}(A)>0\quad\Longleftrightarrow\quad \lambda_1=\lambda_2.$$

The conjecture asserts that indefinite growth of both infection types has positive probability exactly when their intensities are equal, for both the discrete model and the continuum model under the specified condition on $F$. The supplied text gives no resolution, so the conjecture is recorded as open.