Distributional convergence conjecture for final infection sizes

Consider the competing long-range infection model and the notation of the theorem concerning absence of coexistence, including the final-size quantity $M_n$ and the parameter $Z_n$. Let $E_1,E_2$ be independent rate-one exponential random variables. If $c_n=(c+o(1))\log n$ for some $c\in\mathbb R$ and $|c_n|\to\infty$, then

$$M_n n^{-\min\{Z_n,1/Z_n\}}\overset{d}{\longrightarrow}E_1E_2^{-1/(|c|+1)}.$$

If instead $c_n\to c\in\mathbb R$, then

$$\frac{M_n}{n}\overset{d}{\longrightarrow}\frac{E_1}{E_1+E_2\mathrm e^c}.$$

These are proposed stronger distributional results in regimes where coexistence or the relevant case of absence of coexistence occurs. The paper explicitly says that the authors are unable to prove the weaker result in the convergent-$c_n$ regime, so the conjecture remains open.