Optimal thermal clustering distinction between massless and massive theories

Let $F_{A,B}(\boldsymbol{x})$ denote the spatial two-point correlation function of observables $A$ and $B$ at spatial separation $\boldsymbol{x}$ in a thermal relativistic quantum field theory. Write $m=0$ for the massless case and $m>0$ for the massive case. The conjectured optimality of the thermal decay theorem suggests the following bounds:

Optimal thermal clustering conjecture. In the massless case, the decay has the form

$$O\left(|\boldsymbol{x}|^{-\delta}\right)\quad\text{with}\quad \delta\leq 1,$$

whereas in the massive case it has the form

$$O\left(|\boldsymbol{x}|^{-\delta}\right)\quad\text{with}\quad \delta>1.$$

This conjecture would give a sharp distinction between massless and massive thermal relativistic quantum field theories. The massless bound agrees with the $|\boldsymbol{x}|^{-1}$ decay of the massless free field in $3+1$ dimensions, while the corresponding optimal massive bound is expected to decay faster than $|\boldsymbol{x}|^{-1}$; the general claim is left open.