Small-hole stationary-measure convergence conjecture

Let $D_1$ be the torus used in the paper, let $Q_r$ be the stationary distribution of $(X,Y)$ that does not charge the diagonal for the domain obtained by deleting a ball of radius $r$, and let $\operatorname{Unif}((D_1)^2)$ denote the uniform probability distribution on $(D_1)^2$.

Small-hole convergence conjecture. The measures $Q_r$ converge to $\operatorname{Unif}((D_1)^2)$ when $r\to0$.

This conjecture describes the limiting stationary law as the deleted ball shrinks. It is conditional on the preceding stationary-distribution conjecture and is not proved in the supplied text.