The universal profile conjecture for high-dimensional percolation

Let $W^*$ be the Brownian excursion of length $1$, let $\Psi(x)=\mathbb{E}\exp\left[x\int_0^1W^*(t)\,\mathrm dt\right]$ be the moment-generating function of its area, and define

$$F(x,s)=\frac16x^3-\frac{s}{2}x^2+\frac{s^2}{2}x,$$ $$\mathrm d\sigma_s=\frac{1}{\sqrt{2\pi}}x^{-5/2}\Psi(x^{3/2})e^{-F(x,s)}\,\mathrm dx,\qquad f_{\rm perc}(s)=\int_0^\infty x^2\,\mathrm d\sigma_s.$$

On $\mathbb{T}_r$, let $\tau_p^{\mathbb{T}_r}(0,x)=\mathbb{P}_p^{\mathbb{T}_r}(0\xleftrightarrow{}x)$ and $\chi^{\mathbb{T}_r}(p)=\sum_{x\in\mathbb{T}_r}\tau_p^{\mathbb{T}_r}(0,x)$; on $\mathbb{Z}^d$, let $\tau_p$ be the two-point function and $p_c$ the critical value. Universal profile conjecture for percolation. For $d>6$, with window scale $w_r=a_dr^{-d/3}$ for suitably chosen $a_d>0$ and $s\in\mathbb{R}$, as $r\to\infty$,

$$\chi^{\mathbb{T}_r}(p_c+sw_r)\sim \operatorname{const}_d f_{\rm perc}(s)r^{d/3},$$ $$\tau^{\mathbb{T}_r}_{p_c+sw_r}(0,x)-\tau_{p_c}(0,x)\sim \operatorname{const}_d f_{\rm perc}(s)r^{-2d/3}.$$

This predicts a universal finite-size scaling profile for susceptibility and the plateau-like correction to the two-point function in the high-dimensional percolation regime. The source gives no resolution evidence, so the conjecture remains open.