Coexistence conjecture for competing Bernoulli percolation infections

Consider the competition process on $\mathbb{Z}^d$ with blue and yellow infections, whose transmission parameters are $p_{\text{b}}$ and $p_{\text{y}}$, respectively. Let $p_c$ be the critical probability for Bernoulli percolation and let $\overrightarrow{p_c}$ denote the critical probability for oriented percolation. Coexistence conjecture. If

$$p_{\text{b}}=p_{\text{y}}>p_c,$$

then coexistence occurs with positive probability, whereas if $p_{\text{b}}\ne p_{\text{y}}$ and at least one of them is strictly smaller than $\overrightarrow{p_c}$, then coexistence cannot occur. The coexistence statement has already been proved in a previous paper of the authors; the unequal-parameter assertion is the part addressed by the present work.