Truncated-Gaussian distribution 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

$$P_\beta(z;\phi)=\lim_{T\to\infty}\frac{1}{T}\int_0^T H\bigl(z\sigma(\beta;\phi)-v_\beta(t;\phi)\bigr)\,dt,$$

where $H$ is the Heaviside step function. Let $K_0$ be the limit of $K(\beta;\phi)/\sigma(\beta;\phi)$ as $\beta\to\infty$ through $I$, and let $p(v;0,1,K_0)$ denote the density defined in the source, truncated at $\pm K_0$. Assume $-\phi(t)+\sin(2\pi\beta\phi(t))\not\equiv0$. Truncated-Gaussian distribution conjecture. For every such $\phi$ and every $z\in\mathbb{R}$,

$$\lim_{\substack{\beta\in I\beta\to\infty}}P_\beta(z;\phi)=\int_{-\infty}^{z}p(v;0,1,K_0)\,dv.$$

The claim predicts convergence of the normalized time-series distribution to a Gaussian law truncated according to the universal amplitude ratio; it is stated without a proof.