Stationary distribution conjecture for two reflected Brownian motions in dimension three

Let $D$ be the compact domain introduced in the paper, let $(X,Y)$ be the coupled reflected Brownian motion process in dimension $d=3$, and let $r$ be the radius parameter. Write $\{(x,x):x\in\overline D\}$ for the diagonal.

Stationary distribution conjecture. If $d=3$, then there is $r_0>0$ such that for $r\leq r_0$, the process $(X,Y)$ has a stationary distribution $Q$ which does not charge the diagonal $\{(x,x):x\in\overline D\}$. There is only one stationary distribution for $(X,Y)$ which does not charge the diagonal.

The conjecture strengthens the preceding result that, for sufficiently small $r$, the distance between distinct trajectories does not converge to zero. The asserted existence and uniqueness of a stationary law away from the absorbing diagonal are not proved in the supplied text.