Clustering bound for truncated correlations in unbounded spin systems

Let $\psi^{\sharp}$ denote either spin field, let $\lambdaup>0$ be sufficiently small, and let $\Lambdaup\subset\mathbb{Z}^d$ be a finite volume. For even $n\geq 2$, write $\langle\psi^{\sharp_1}(\mathbf{0}),\psi^{\sharp_2}(\mathbf{x}_2),\ldots,\psi^{\sharp_n}(\mathbf{x}_n)\rangle_{\Lambdaup,\lambdaup}^{\mathrm{T}}$ for the truncated $n$-point correlation, with each $\sharp_i$ an arbitrary assignment of the available sharp symbols. Clustering bound conjecture. There exists a constant $c(J)>0$ such that, for $\lambdaup>0$ small enough, every even integer $n\geq 2$, every finite volume $\Lambdaup\subset\mathbb{Z}^d$, and every assignment of the $\sharp$ symbols satisfy

$$\sum_{\mathbf{x}_2,\ldots,\mathbf{x}_n\in\Lambdaup}\left|\langle\psi^{\sharp_1}(\mathbf{0}),\psi^{\sharp_2}(\mathbf{x}_2),\ldots,\psi^{\sharp_n}(\mathbf{x}_n)\rangle_{\Lambdaup,\lambdaup}^{\mathrm{T}}\right|\leq c(J)^n\lambdaup^{\frac{n}{2}-1}n!.$$

This conjecture would provide a uniform $l^1$-clustering estimate with factorial growth in the number of fields, extending the tree-level bound to the full truncated correlations. The supplied text does not state whether the conjecture has been proved or disproved.