Negative association conjecture for the arboreal gas

Let $\beta>0$, and for distinct edges write $\mathbb{P}_{\beta}[e_1,\ldots,e_k]$ for the probability that all listed edges occur in the forest. Negative association conjecture. For distinct edges $ij$ and $kl$,

$$\mathbb{P}_{\beta}[ij,kl]\leq \mathbb{P}_{\beta}[ij]\mathbb{P}_{\beta}[kl].$$

More generally, for all distinct edges $i_1j_1,\ldots,i_nj_n$ and $m<n$,

$$\mathbb{P}_{\beta}[i_1j_1,\ldots,i_nj_n]\leq \mathbb{P}_{\beta}[i_1j_1,\ldots,i_mj_m]\,\mathbb{P}_{\beta}[i_{m+1}j_{m+1},\ldots,i_nj_n].$$

The conjecture was previously stated in the cited literature and is expected by the authors to extend to general positive edge weights; no resolution is supplied here.