Zero-mean conjecture for irregular delay-equation solutions

Let $I$ be the parameter interval for which $\beta$ yields no stable steady state or stable periodic solution, and let $v_\beta(t;\phi)$ solve

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

For an initial function $\phi\in C([-1,0],\mathbb{R})$ satisfying $-\phi(t)+\sin(2\pi\beta\phi(t))\not\equiv0$, define

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

Zero-mean conjecture. For every such $\phi$ and every $\beta\in I$, $\mu(\beta;\phi)=0$. The authors state that this is unproved for irregular solutions; it holds for symmetric periodic solutions, and the conjecture remains open.