Suppose $D$ is a bounded Lipschitz domain in $R^n$ and $X_1$, $X_2$ are two independent reflected Brownian motions (RBMs) in $D$. Is it true that

$$P[X_1(t)\neq X_2(t) \text{ for all }t>0]=1$$

It seems to be true for dimension $n\geq 2$.

But how to prove it? Does anyone know any reference? Thanks!