Many-hole domain conjecture for reflected Brownian motions

For $j=1,\dots,k$, let ${\mathcal B}(x_j,r)\subset {\mathcal B}(0,1)$, and define

$$D_2={\mathcal B}(0,1)\setminus\bigcup_{j=1}^k\overline{{\mathcal B}(x_j,r)}\subset\mathbb R^3.$$

Let Theorem j13.5 denote the paper's assertion that, in dimension three and for sufficiently small $r$, distinct reflected Brownian motions have distance with positive limsup.

Many-hole domain conjecture. If $k$ is sufficiently large and

$$\frac{\min_{1\leq j\leq k}(1-|x_j|)+\min_{1\leq i<j\leq k}|x_i-x_j|}{r}$$

is sufficiently large, then Theorem j13.5 holds for $D_2$.

The conjecture proposes an extension of the three-dimensional torus result to certain bounded subsets of $\mathbb R^3$ containing many well-separated deleted balls. The supplied text does not establish this extension.