High-temperature exponential clustering conjecture for short-range interactions

Let $C_\mathrm{int}>0$, $b>0$, and $\nu\in\mathbb{N}$. For an interaction $\varPsi$ on $\mathbb{Z}^\nu$, assume

$$\norm{\varPsi}_{\exp(-b\,\cdot)}<C_\mathrm{int}.$$

High-temperature exponential clustering conjecture. There exist $\beta^*$, $C_\mathrm{Cov}$, $c_\mathrm{Cov}>0$, and $n\in\mathbb{N}$ such that, for every $\beta<\beta^*$ and every finite $\varLambda\Subset\mathbb{Z}^\nu$, the Gibbs state $\rho^\varLambda_\beta$ satisfies decay of correlations, with

$$\operatorname{Cov}_{\rho^\varLambda_\beta}(X;Y) \leq C_\mathrm{Cov}\,\abs{X}^n\,\abs{Y}^n\,\mathrm{e}^{-c_\mathrm{Cov}\dist{X,Y}}$$

for all $X,Y\subset\varLambda$. This is the expected extension of the known high-temperature exponential decay for finite-range systems to short-range interactions in arbitrary dimension; the conjectural point is the uniform bound for interactions with exponentially decaying tails.