Brownian large-scale limit for the one-dimensional global solution

In one spatial dimension, let $\Phi$ be the global solution with $a=b=0$, and regard it modulo additive constants so that it has stationary spatial increments. Brownian limit conjecture. There is a constant $\sigma>0$ such that the rescaled process

$$\left(\frac{\Phi(0,sx)}{\sigma\sqrt{s}}\right)_{x\in\mathbb{R}}$$

converges in distribution, as $s\to+\infty$, to the standard two-sided Wiener process. This predicts diffusive spatial fluctuations of the global solution and is presented as a one-dimensional consequence of the expected large-scale behaviour; no proof or resolution is given.