Coexistence conjecture at the critical long-range exponent

Consider the competing long-range infection model on $\mathbb T_n^d$ with long-range parameters $\alpha_\ominus(n)$ and $\alpha_\oplus(n)$, infection-rate ratio $\lambda(n)$, and coexistence of the two infections. Critical long-range coexistence conjecture. If

$$\alpha_\ominus(n)=\alpha_\oplus(n)=d$$

for every $n\in\mathbb N$ and $\lambda(n)=\lambda>0$ is fixed, then coexistence occurs if and only if $\lambda=1$. This is presented as a first step toward the unresolved regime in which one or both long-range parameters can reach $d$; the source gives no resolution.