The computational hardness threshold conjecture for weighted independent sets

Let $d\geq 3$ be the maximum degree, let $\ell$ be the activity, and let $\ell_c(d)=(d-1)^{d-1}/(d-2)^d$ be the critical activity for decay of correlations on the infinite $d$-regular tree. Consider the weighted sum of independent sets, with each independent set $I$ having weight $\ell^{|I|}$. Computational hardness threshold conjecture. For every $d>3$ and all $\ell>\ell_c(d)$, unless $RP=NP$, there does not exist a fully polynomial approximation scheme for counting weighted independent sets in graphs of maximum degree $d$ with activity $\ell$. This conjecture identifies the computational threshold with the uniqueness threshold; below $\ell_c(d)$ a fully polynomial approximation scheme is known, while hardness above the threshold remains open.