Variance-scaling conjecture for deterministic Brownian motion

Let $I$ be the parameter interval for which the delay equation has no stable steady state or stable periodic solution. For $\phi\in C([-1,0],\mathbb{R})$, let $v_\beta(t;\phi)$ solve

$$\frac{d v}{d t}=-v+\sin(2\pi\beta v(t-1)),\qquad v(t)=\phi(t)\quad(-1\leq t\leq0),$$

and define

$$\sigma(\beta;\phi)=\lim_{T\to\infty}\sqrt{\frac{1}{T}\int_0^T v_\beta(t;\phi)^2\,dt}.$$

Assume $-\phi(t)+\sin(2\pi\beta\phi(t))\not\equiv0$. Variance-scaling conjecture. The limit

$$\lim_{\substack{\beta\in I\beta\to\infty}}\beta^{1/2}\sigma(\beta;\phi)$$

exists, is independent of $\phi$, and is positive. This predicts universal $\beta^{-1/2}$ scaling of the long-time root-mean-square amplitude, but remains unproved.