Near-critical massive Gaussian free field conjecture for the high-dimensional Ising model

Consider the near-critical Ising model on $a\mathbb{Z}^d$ with $d>4$, at $\beta=\beta_c$, with external field $H=a^{(d+2)/2}h$ for some $h>0$. Let $\delta_x$ denote the unit Dirac point measure at $x$, and define

$$\Phi^{a,h}:=a^{(d+2)/2}\sum_{x\in a\mathbb{Z}^d}[\sigma_x-\langle\sigma_x\rangle_{\mathbb{Z}^d,H}]\delta_x.$$

Near-critical scaling-limit conjecture. As $a\downarrow0$,

$$\Phi^{a,h}\Longrightarrow \text{massive Gaussian free field on }\mathbb{R}^d.$$

Here $\Longrightarrow$ denotes convergence in distribution. The covariance of the limiting field is proportional to the kernel of $(-\Delta+m^2)^{-1}$ for some $m>0$, and therefore decays exponentially. This is the proposed infinite-volume near-critical scaling limit in dimensions above four; the supplied passage gives no resolution status.