Goldstone-mode decay conjecture for the arboreal gas

Let the arboreal gas be defined on $\mathbb{Z}^d$ with $d\geq 3$, and write $\mathbb{P}_{\beta}[i\leftrightarrow j]$ for the probability that $i$ and $j$ lie in the same tree. Goldstone-mode decay conjecture. When $\beta>\beta_c$,

$$\mathbb{P}_{\beta}[i\leftrightarrow j]-c_\beta\approx |i-j|^{-(d-2)}\qquad\text{as }|i-j|\to\infty.$$

This conjecture concerns the long-range correction to the positive connectivity in the supercritical phase and is presented as a distinction between the arboreal gas and ordinary percolation; no proof or resolution is given in the source.