Contractible-domain conjecture for shy co-adapted couplings

Let $D$ be a bounded contractible domain in a Euclidean space of arbitrary dimension, and consider reflecting Brownian motions in $D$ together with co-adapted couplings. A coupling is shy if the two coupled motions remain separated in the sense defined in the paper. Contractible-domain conjecture. There can be no shy co-adapted coupling for reflecting Brownian motions in bounded contractible domains in any dimension. The planar case follows from the equivalence of the CAT$(0)$ condition and simple-connectedness, but the conjecture in higher dimensions remains open; the authors indicate that the star-shaped case may require mainly technical work, whereas substantial progress in general will require new ideas.