The bouncing-ray conjecture for tree-indexed random walks

Consider a tree-indexed random walk on $\mathbb{R}$ with increment distribution $\mu$. A ray $x=(v_0,v_1,\ldots)\in\partial\Gamma$ is an escaping ray if $S(v_n)\to\infty$ as $n\to\infty$, and a bouncing ray if

$$\infty>\liminf_{n\to\infty}S(v_n)>-\infty.$$

Bouncing-ray conjecture. If $\mu$ has mean zero and finite variance and $\Gamma$ almost surely has bouncing rays, then $\Gamma$ almost surely has escaping rays. This is presented as an open question about tree-indexed random walks; the source gives no general resolution.