Arboreal gas phase-transition conjecture in dimensions at least three

Let the arboreal gas be defined on $\mathbb{Z}^d$ with $d\geq 3$, let $B_r$ be the radius-$r$ ball, and let $T_0$ denote the tree containing the origin. Write $\mathbb{P}_{\beta}[i\leftrightarrow j]$ for the probability that $i$ and $j$ lie in the same tree, and let $\beta$ be the parameter of the model. Arboreal gas phase-transition conjecture. There exists $\beta_c>0$ such that

$$\lim_{r\to\infty}\mathbb{E}_{\beta}\frac{|T_0\cap B_r|}{|B_r|}=\begin{cases}0&\text{if }\beta<\beta_c,\\>0&\text{if }\beta>\beta_c.\end{cases}$$

Moreover, when $\beta<\beta_c$, there is $c_\beta>0$ such that

$$\mathbb{P}_{\beta}[i\leftrightarrow j]\leq Ce^{-c_\beta|i-j|}\qquad(i,j\in\mathbb{Z}^d),$$

while when $\beta>\beta_c$,

$$\mathbb{P}_{\beta}[i\leftrightarrow j]\geq c_\beta>0.$$

The first equality for sufficiently small $\beta$ is known, but the existence of a percolating phase for large $\beta$ and the existence of a sharp critical value remain open; the conjecture predicts mean-field-type phase-transition behaviour.