The hard-core uniqueness conjecture at the tree threshold

Let $G$ be a countably infinite graph of maximum degree

, with vertex fugacities $w_x$. **The hard-core uniqueness conjecture.** For some

, the hard-core lattice gas on $G$ has a unique Gibbs measure whenever

$$0 \le w_x \le (1-\epsilon)\frac{(\Delta-1)^{\Delta-1}}{(\Delta-2)^\Delta}$$

for all vertices $x$; perhaps this remains true with $\epsilon=0$. This would improve the known uniqueness bounds, which are of order $1/\Delta$, to the conjectured sharp tree-scale threshold, but the statement is unresolved in the source.