Equality of the critical thresholds for three-dimensional Brownian loop soup

Let $\lpha_c$, $\lpha_u$, $\lpha_{\rm tr}$, and $\lpha_{\mathop{\rm b2b}}$ denote the thresholds defined in the paper for, respectively, percolation, uniqueness, truncated-model percolation, and the remaining cutoff-based phase transition. Threshold equality conjecture.

$$\alpha_c=\alpha_u=\alpha_{\rm tr}=\alpha_{\mathop{\rm b2b}}.$$

The paper states that all these thresholds are believed to coincide, but this remains an open problem. In particular, proving $\lpha_c=\alpha_u$ would rule out a phase with infinitely many unbounded clusters, each dense in $\mathbb{R}^3$; a naive Burton--Keane argument does not apply because infinitely many small loops almost surely intersect the boundary of any given ball.