Competition conjecture on the long-range percolation cluster

Consider the long-range percolation graph on $\mathbb T_n^d$ with all nearest-neighbour edges and each non-nearest-neighbour edge included independently with probability $\|x-y\|^{-\alpha}$, where $\alpha\in(0,d)$. Start competing first-passage percolation from two uniformly chosen vertices, with the $\ominus$ infection spreading at rate $1$ and the $\oplus$ infection at rate $\lambda>0$. Let $M_n$ denote the size of the smaller infection when all vertices have been infected. Long-range percolation competition conjecture. If $\lambda=1$, coexistence occurs. If $\lambda\ne1$, then

$$\frac{\log M_n}{\log n}\overset{\mathbb P}{\longrightarrow}\min\{\lambda,1/\lambda\}.$$

The claim is motivated by the expectation that the long-range percolation cluster has behavior similar to the mean-field model studied in the paper; no proof or disproof is supplied.